GNU MP

The GNU Multiple Precision Arithmetic Library Edition 4.1 24 May 2002

by Torbjörn Granlund, Swox AB

tege@swox.com

This manual describes how to install and use the GNU multiple precision arithmetic library, version 4.1. Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002 Free Software Foundation, Inc. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.1 or any later version published by the Free Software Foundation; with no Invariant Sections, with the Front-Cover Texts being “A GNU Manual”, and with the Back-Cover Texts being “You have freedom to copy and modify this GNU Manual, like GNU software”. A copy of the license is included in Appendix C [GNU Free Documentation License], page 111.

Table of Contents

GNU MP Copying Conditions . . . . . . . . . . . . . . . . . . . . 1

1 Introduction to GNU MP . . . . . . . . . . . . . . . . . . . . . 2

1.1 How to use this Manual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Installing GMP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Build Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 ABI and ISA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Notes for Package Builds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Notes for Particular Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Known Build Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 GMP Basics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1 Headers and Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Nomenclature and Types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 Function Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.4 Variable Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.5 Parameter Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.6 Memory Management. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.7 Reentrancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.8 Useful Macros and Constants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.9 Compatibility with older versions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.10 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.11 Debugging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.12 Profiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.13 Autoconf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Reporting Bugs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Integer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.1 Initialization Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.2 Assignment Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.3 Combined Initialization and Assignment Functions . . . . . . . . . . . . 27 5.4 Conversion Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.5 Arithmetic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.6 Division Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.7 Exponentiation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.8 Root Extraction Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.9 Number Theoretic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.10 Comparison Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.11 Logical and Bit Manipulation Functions. . . . . . . . . . . . . . . . . . . . . 34 5.12 Input and Output Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.13 Random Number Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.14 Integer Import and Export . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.15 Miscellaneous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6 Rational Number Functions . . . . . . . . . . . . . . . . . . 39

6.1 Initialization and Assignment Functions . . . . . . . . . . . . . . . . . . . . . . 39 6.2 Conversion Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.3 Arithmetic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.4 Comparison Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.5 Applying Integer Functions to Rationals. . . . . . . . . . . . . . . . . . . . . . 41 6.6 Input and Output Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

7 Floating-point Functions . . . . . . . . . . . . . . . . . . . . . 43

7.1 Initialization Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 7.2 Assignment Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 7.3 Combined Initialization and Assignment Functions . . . . . . . . . . . . 45 7.4 Conversion Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7.5 Arithmetic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 7.6 Comparison Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7.7 Input and Output Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7.8 Miscellaneous Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

8 Low-level Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 50

8.1 Nails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

9 Random Number Functions . . . . . . . . . . . . . . . . . . 58

9.1 Random State Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 9.2 Random State Seeding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

10 Formatted Output . . . . . . . . . . . . . . . . . . . . . . . . . . 60

10.1 Format Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 10.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 10.3 C++ Formatted Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

11 Formatted Input . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

11.1 Formatted Input Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 11.2 Formatted Input Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 11.3 C++ Formatted Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

12 C++ Class Interface . . . . . . . . . . . . . . . . . . . . . . . . . 68

12.1 C++ Interface General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 12.2 C++ Interface Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 12.3 C++ Interface Rationals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 12.4 C++ Interface Floats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 12.5 C++ Interface MPFR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 12.6 C++ Interface Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 12.7 C++ Interface Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

13 Berkeley MP Compatible Functions . . . . . . . . . 76

14 Custom Allocation. . . . . . . . . . . . . . . . . . . . . . . . . . 78

15 Language Bindings . . . . . . . . . . . . . . . . . . . . . . . . . 80 16 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

16.1 Multiplication. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 16.1.1 Basecase Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 16.1.2 Karatsuba Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . 83 16.1.3 Toom-Cook 3-Way Multiplication . . . . . . . . . . . . . . . . . . 84 16.1.4 FFT Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 16.1.5 Other Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

16.2 Division Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 16.2.1 Single Limb Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 16.2.2 Basecase Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 16.2.3 Divide and Conquer Division . . . . . . . . . . . . . . . . . . . . . . 89 16.2.4 Exact Division. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 16.2.5 Exact Remainder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 16.2.6 Small Quotient Division . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

16.3 Greatest Common Divisor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 16.3.1 Binary GCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 16.3.2 Accelerated GCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 16.3.3 Extended GCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 16.3.4 Jacobi Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

16.4 Powering Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 16.4.1 Normal Powering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 16.4.2 Modular Powering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

16.5 Root Extraction Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 16.5.1 Square Root. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 16.5.2 Nth Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 16.5.3 Perfect Square. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 16.5.4 Perfect Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

16.6 Radix Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 16.6.1 Binary to Radix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 16.6.2 Radix to Binary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

16.7 Other Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 16.7.1 Factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 16.7.2 Binomial Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 16.7.3 Fibonacci Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 16.7.4 Lucas Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

16.8 Assembler Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 16.8.1 Code Organisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 16.8.2 Assembler Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 16.8.3 Carry Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 16.8.4 Cache Handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 16.8.5 Floating Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 16.8.6 SIMD Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 16.8.7 Software Pipelining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 16.8.8 Loop Unrolling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

17 Internals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

17.1 Integer Internals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 17.2 Rational Internals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 17.3 Float Internals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 17.4 Raw Output Internals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 17.5 C++ Interface Internals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Appendix A Contributors. . . . . . . . . . . . . . . . . . . . . 107

Appendix B References . . . . . . . . . . . . . . . . . . . . . . . 109

B.1 Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 B.2 Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 iv GNU MP 4.1

Appendix C GNU Free Documentation License . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

C.1 ADDENDUM: How to use this License for your documents… 116

Concept Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Function and Type Index. . . . . . . . . . . . . . . . . . . . . . . 119

GNU MP Copying Conditions

This library is free; this means that everyone is free to use it and free to redistribute it on a free basis. The library is not in the public domain; it is copyrighted and there are restrictions on its distribution, but these restrictions are designed to permit everything that a good cooperating citizen would want to do. What is not allowed is to try to prevent others from further sharing any version of this library that they might get from you.

Specifically, we want to make sure that you have the right to give away copies of the library, that you receive source code or else can get it if you want it, that you can change this library or use pieces of it in new free programs, and that you know you can do these things.

To make sure that everyone has such rights, we have to forbid you to deprive anyone else of these rights. For example, if you distribute copies of the GNU MP library, you must give the recipients all the rights that you have. You must make sure that they, too, receive or can get the source code. And you must tell them their rights.

Also, for our own protection, we must make certain that everyone finds out that there is no warranty for the GNU MP library. If it is modified by someone else and passed on, we want their recipients to know that what they have is not what we distributed, so that any problems introduced by others will not reflect on our reputation.

The precise conditions of the license for the GNU MP library are found in the Lesser General Public License version 2.1 that accompanies the source code, see ‘COPYING.LIB’. Certain demonstration programs are provided under the terms of the plain General Public License version 2,

see ‘COPYING’.

1 Introduction to GNU MP

GNU MP is a portable library written in C for arbitrary precision arithmetic on integers, rational numbers, and floating-point numbers. It aims to provide the fastest possible arithmetic for all applications that need higher precision than is directly supported by the basic C types.

Many applications use just a few hundred bits of precision; but some applications may need thousands or even millions of bits. GMP is designed to give good performance for both, by choosing algorithms based on the sizes of the operands, and by carefully keeping the overhead at a minimum.

The speed of GMP is achieved by using fullwords as the basic arithmetic type, by using sophisticated algorithms, by including carefully optimized assembly code for the most common inner loops for many different CPUs, and by a general emphasis on speed (as opposed to simplicity or elegance).

There is carefully optimized assembly code for these CPUs: ARM, DEC Alpha 21064, 21164, and 21264, AMD 29000, AMD K6, K6-2 and Athlon, Hitachi SuperH and SH-2, HPPA 1.0, 1.1 and 2.0, Intel Pentium, Pentium Pro/II/III, Pentium 4, generic x86, Intel IA-64, i960, Motorola MC68000, MC68020, MC88100, and MC88110, Motorola/IBM PowerPC 32 and 64, National NS32000, IBM POWER, MIPS R3000, R4000, SPARCv7, SuperSPARC, generic SPARCv8, UltraSPARC, DEC VAX, and Zilog Z8000. Some optimizations also for Cray vector systems, Clipper, IBM ROMP (RT), and Pyramid AP/XP.

There is a mailing list for GMP users. To join it, send a mail to gmp-request@swox.com with the word ‘subscribe’ in the message body (not in the subject line).

For up-to-date information on GMP, please see the GMP web pages at

     http://swox.com/gmp/

The latest version of the library is available at

     ftp://ftp.gnu.org/gnu/gmp

Many sites around the world mirror ‘ftp.gnu.org’, please use a mirror near you, see

http://www.gnu.org/order/ftp.html for a full list.

1.1 How to use this Manual

Everyone should read Chapter 3 [GMP Basics], page 14. If you need to install the library yourself, then read Chapter 2 [Installing GMP], page 3. If you have a system with multiple ABIs, then read Section 2.2 [ABI and ISA], page 8, for the compiler options that must be used on applications.

The rest of the manual can be used for later reference, although it is probably a good idea to glance through it.

2 Installing GMP

GMP has an autoconf/automake/libtool based configuration system. On a Unix-like system a basic build can be done with

./configure
make

Some self-tests can be run with

make check

And you can install (under ‘/usr/local’ by default) with

make install

If you experience problems, please report them to bug-gmp@gnu.org. See Chapter 4 [Reporting Bugs], page 24, for information on what to include in useful bug reports.

2.1 Build Options

All the usual autoconf configure options are available, run ‘./configure --help’ for a summary. The file ‘INSTALL.autoconf’ has some generic installation information too.

Non-Unix Systems ‘configure’ requires various Unix-like tools. On an MS-DOS system DJGPP can be used, and on MS Windows Cygwin or MINGW can be used,

     http://www.cygnus.com/cygwin
     http://www.delorie.com/djgpp
     http://www.mingw.org

The ‘macos’ directory contains an unsupported port to MacOS 9 on Power Macintosh, see ‘macos/README’. Note that MacOS X “Darwin” should use the normal Unix-style ‘./configure’. It might be possible to build without the help of ‘configure’, certainly all the code is there, but unfortunately you’ll be on your own.

Build Directory To compile in a separate build directory, cd to that directory, and prefix the configure command with the path to the GMP source directory. For example

     cd /my/build/dir
     /my/sources/gmp-4.1/configure

Not all ‘make’ programs have the necessary features (VPATH) to support this. In particular, SunOS and Slowaris make have bugs that make them unable to build in a separate directory. Use GNU make instead.

‘--disable-shared’, ‘--disable-static’

By default both shared and static libraries are built (where possible), but one or other can be disabled. Shared libraries result in smaller executables and permit code sharing between separate running processes, but on some CPUs are slightly slower, having a small cost on each function call.

Native Compilation, ‘--build=CPU-VENDOR-OS’

For normal native compilation, the system can be specified with ‘--build’. By default ‘./configure’ uses the output from running ‘./config.guess’. On some systems ‘./config.guess’ can determine the exact CPU type, on others it will be necessary to give it explicitly. For example,

     ./configure --build=ultrasparc-sun-solaris2.7

In all cases the ‘OS’ part is important, since it controls how libtool generates shared libraries. Running ‘./config.guess’ is the simplest way to see what it should be, if you don’t know already.

Cross Compilation, ‘--host=CPU-VENDOR-OS’

When cross-compiling, the system used for compiling is given by ‘--build’ and the system where the library will run is given by ‘--host’. For example when using a FreeBSD Athlon system to build GNU/Linux m68k binaries,

     ./configure --build=athlon-pc-freebsd3.5 --host=m68k-mac-linux-gnu

Compiler tools are sought first with the host system type as a prefix. For example m68k-mac-linux-gnu-ranlib is tried, then plain ranlib. This makes it possible for a set of cross-compiling tools to co-exist with native tools. The prefix is the argument to ‘--host’, and this can be an alias, such as ‘m68k-linux’. But note that tools don’t have to be setup this way, it’s enough to just have a PATH with a suitable cross-compiling cc etc. Compiling for a different CPU in the same family as the build system is a form of cross-compilation, though very possibly this would merely be special options on a native compiler. In any case ‘./configure’ avoids depending on being able to run code on the build system, which is important when creating binaries for a newer CPU since they very possibly won’t run on the build system. In all cases the compiler must be able to produce an executable (of whatever format) from a standard C main. Although only object files will go to make up ‘libgmp’, ‘./configure’ uses linking tests for various purposes, such as determining what functions are available on the host system. Currently a warning is given unless an explicit ‘--build’ is used when crosscompiling, because it may not be possible to correctly guess the build system type if the PATH has only a cross-compiling cc. Note that the ‘--target’ option is not appropriate for GMP. It’s for use when building compiler tools, with ‘--host’ being where they will run, and ‘--target’ what they’ll produce code for. Ordinary programs or libraries like GMP are only interested in the ‘--host’ part, being where they’ll run. (Some past versions of GMP used ‘--target’ incorrectly.)

CPU types In general, if you want a library that runs as fast as possible, you should configure GMP for the exact CPU type your system uses. However, this may mean the binaries won’t run on older members of the family, and might run slower on other members, older or newer. The best idea is always to build GMP for the exact machine type you intend to run it on. The following CPUs have specific support. See ‘configure.in’ for details of what code and compiler options they select.

  • Alpha: ‘alpha’, ‘alphaev5’, ‘alphaev56’, ‘alphapca56’, ‘alphapca57’,
    ‘alphaev6’, ‘alphaev67’, ‘alphaev68’

• Cray: ‘c90’, ‘j90’, ‘t90’, ‘sv1’

  • HPPA: ‘hppa1.0’, ‘hppa1.1’, ‘hppa2.0’, ‘hppa2.0n’, ‘hppa2.0w’
  • MIPS: ‘mips’, ‘mips3’, ‘mips64’
  • Motorola: ‘m68k’, ‘m68000’, ‘m68010’, ‘m68020’, ‘m68030’, ‘m68040’, ‘m68060’,
    ‘m68302’, ‘m68360’, ‘m88k’, ‘m88110’
  • POWER: ‘power’, ‘power1’, ‘power2’, ‘power2sc’
  • PowerPC: ‘powerpc’, ‘powerpc64’, ‘powerpc401’, ‘powerpc403’, ‘powerpc405’,
    ‘powerpc505’, ‘powerpc601’, ‘powerpc602’, ‘powerpc603’, ‘powerpc603e’,

    ‘powerpc604’, ‘powerpc604e’, ‘powerpc620’, ‘powerpc630’, ‘powerpc740’,
    ‘powerpc7400’, ‘powerpc7450’, ‘powerpc750’, ‘powerpc801’, ‘powerpc821’,
    ‘powerpc823’, ‘powerpc860’,
  • SPARC: ‘sparc’, ‘sparcv8’, ‘microsparc’, ‘supersparc’, ‘sparcv9’,
    ‘ultrasparc’, ‘ultrasparc2’, ‘ultrasparc2i’, ‘ultrasparc3’, ‘sparc64’
  • 80x86 family: ‘i386’, ‘i486’, ‘i586’, ‘pentium’, ‘pentiummmx’, ‘pentiumpro’,
    ‘pentium2’, ‘pentium3’, ‘pentium4’, ‘k6’, ‘k62’, ‘k63’, ‘athlon’
  • Other: ‘a29k’, ‘arm’, ‘clipper’, ‘i960’, ‘ns32k’, ‘pyramid’, ‘sh’, ‘sh2’, ‘vax’,
    ‘z8k’

CPUs not listed will use generic C code.

Generic C Build If some of the assembly code causes problems, or if otherwise desired, the generic C code can be selected with CPU ‘none’. For example,

     ./configure --host=none-unknown-freebsd3.5

Note that this will run quite slowly, but it should be portable and should at least make it possible to get something running if all else fails.

‘ABI’

On some systems GMP supports multiple ABIs (application binary interfaces), meaning data type sizes and calling conventions. By default GMP chooses the best ABI available, but a particular ABI can be selected. For example

     ./configure --host=mips64-sgi-irix6 ABI=n32

See Section 2.2 [ABI and ISA], page 8, for the available choices on relevant CPUs, and what applications need to do.

‘CC’, ‘CFLAGS’

By default the C compiler used is chosen from among some likely candidates, with gcc normally preferred if it’s present. The usual ‘CC=whatever’ can be passed to ‘./configure’ to choose something different. For some systems, default compiler flags are set based on the CPU and compiler. The usual ‘CFLAGS="-whatever"’ can be passed to ‘./configure’ to use something different or to set good flags for systems GMP doesn’t otherwise know. The ‘CC’ and ‘CFLAGS’ used are printed during ‘./configure’, and can be found in each generated ‘Makefile’. This is the easiest way to check the defaults when considering changing or adding something. Note that when ‘CC’ and ‘CFLAGS’ are specified on a system supporting multiple ABIs it’s important to give an explicit ‘ABI=whatever’, since GMP can’t determine the ABI just from the flags and won’t be able to select the correct assembler code. If just ‘CC’ is selected then normal default ‘CFLAGS’ for that compiler will be used (if GMP recognises it). For example ‘CC=gcc’ can be used to force the use of GCC, with default flags (and default ABI).

‘CPPFLAGS’

Any flags like ‘-D’ defines or ‘-I’ includes required by the preprocessor should be set in ‘CPPFLAGS’ rather than ‘CFLAGS’. Compiling is done with both ‘CPPFLAGS’ and ‘CFLAGS’, but preprocessing uses just ‘CPPFLAGS’. This distinction is because most preprocessors won’t accept all the flags the compiler does. Preprocessing is done separately in some configure tests, and in the ‘ansi2knr’ support for K&R compilers.

C++ Support, ‘--enable-cxx’

C++ support in GMP can be enabled with ‘--enable-cxx’, in which case a C++ compiler will be required. As a convenience ‘--enable-cxx=detect’ can be used to enable C++ support only if a compiler can be found. The C++ support consists of a library ‘libgmpxx.la’ and header file ‘gmpxx.h’. A separate ‘libgmpxx.la’ has been adopted rather than having C++ objects within ‘libgmp.la’ in order to ensure dynamic linked C programs aren’t bloated by a dependency on the C++ standard library, and to avoid any chance that the C++ compiler could be required when linking plain C programs. ‘libgmpxx.la’ will use certain internals from ‘libgmp.la’ and can only be expected to work with ‘libgmp.la’ from the same GMP version. Future changes to the relevant internals will be accompanied by renaming, so a mismatch will cause unresolved symbols rather than perhaps mysterious misbehaviour. In general ‘libgmpxx.la’ will be usable only with the C++ compiler that built it, since name mangling and runtime support are usually incompatible between different compilers.

‘CXX’, ‘CXXFLAGS’

When C++ support is enabled, the C++ compiler and its flags can be set with variables ‘CXX’ and ‘CXXFLAGS’ in the usual way. The default for ‘CXX’ is the first compiler that works from a list of likely candidates, with g++ normally preferred when available. The default for ‘CXXFLAGS’ is to try ‘CFLAGS’, ‘CFLAGS’ without ‘-g’, then for g++ either ‘-g -O2’ or ‘-O2’, or for other compilers ‘-g’ or nothing. Trying ‘CFLAGS’ this way is convenient when using ‘gcc’ and ‘g++’ together, since the flags for ‘gcc’ will usually suit ‘g++’. It’s important that the C and C++ compilers match, meaning their startup and runtime support routines are compatible and that they generate code in the same ABI (if there’s a choice of ABIs on the system). ‘./configure’ isn’t currently able to check these things very well itself, so for that reason ‘--disable-cxx’ is the default, to avoid a build failure due to a compiler mismatch. Perhaps this will change in the future. Incidentally, it’s normally not good enough to set ‘CXX’ to the same as ‘CC’. Although gcc for instance recognises ‘foo.cc’ as C++ code, only g++ will invoke the linker the right way when building an executable or shared library from object files.

Temporary Memory, ‘--enable-alloca=<choice>’

GMP allocates temporary workspace using one of the following three methods, which

can be selected with for instance ‘--enable-alloca=malloc-reentrant’.

• ‘alloca’ - C library or compiler builtin.

• ‘malloc-reentrant’ - the heap, in a re-entrant fashion.

• ‘malloc-notreentrant’ - the heap, with global variables.

For convenience, the following choices are also available. ‘--disable-alloca’ is the

same as ‘--enable-alloca=no’.

• ‘yes’ - a synonym for ‘alloca’.

  • ‘no’ - a synonym for ‘malloc-reentrant’.

• ‘reentrant’ - alloca if available, otherwise ‘malloc-reentrant’. This is the default.

  • ‘notreentrant’ - alloca if available, otherwise ‘malloc-notreentrant’.

alloca is reentrant and fast, and is recommended, but when working with large numbers it can overflow the available stack space, in which case one of the two malloc methods will need to be used. Alternately it might be possible to increase available stack with limit, ulimit or setrlimit, or under DJGPP with stubedit or _stklen. Note that depending on the system the only indication of stack overflow might be a segmentation violation. ‘malloc-reentrant’ is, as the name suggests, reentrant and thread safe, but ‘malloc-notreentrant’ is faster and should be used if reentrancy is not required.

The two malloc methods in fact use the memory allocation functions selected by mp_ set_memory_functions, these being malloc and friends by default. See Chapter 14 [Custom Allocation], page 78.

An additional choice ‘--enable-alloca=debug’ is available, to help when debugging memory related problems (see Section 3.11 [Debugging], page 20).

FFT Multiplication, ‘--disable-fft’ By default multiplications are done using Karatsuba, 3-way Toom-Cook, and Fermat FFT. The FFT is only used on large to very large operands and can be disabled to save code size if desired.

Berkeley MP, ‘--enable-mpbsd’

The Berkeley MP compatibility library (‘libmp’) and header file (‘mp.h’) are built and installed only if ‘--enable-mpbsd’ is used. See Chapter 13 [BSD Compatible Functions], page 76.

MPFR, ‘--enable-mpfr’

The optional MPFR functions are built and installed only if ‘--enable-mpfr’ is used. These are in a separate library ‘libmpfr.a’ and are documented separately too (see section “Introduction to MPFR” in MPFR).

Assertion Checking, ‘--enable-assert’ This option enables some consistency checking within the library. This can be of use while debugging, see Section 3.11 [Debugging], page 20.

Execution Profiling, ‘--enable-profiling=prof/gprof’

Profiling support can be enabled either for prof or gprof. This adds ‘-p’ or ‘-pg’ respectively to ‘CFLAGS’, and for some systems adds corresponding mcount calls to the assembler code. See Section 3.12 [Profiling], page 22.

‘MPN_PATH’

Various assembler versions of each mpn subroutines are provided. For a given CPU, a search is made though a path to choose a version of each. For example ‘sparcv8’ has

     MPN_PATH="sparc32/v8 sparc32 generic"

which means look first for v8 code, then plain sparc32 (which is v7), and finally fall back on generic C. Knowledgeable users with special requirements can specify a different path. Normally this is completely unnecessary.

Demonstration Programs The ‘demos’ subdirectory has some sample programs using GMP. These aren’t built or installed, but there’s a ‘Makefile’ with rules for them. For instance,

     make pexpr
     ./pexpr 68^975+10

Documentation The document you’re now reading is ‘gmp.texi’. The usual automake targets are available to make PostScript ‘gmp.ps’ and/or DVI ‘gmp.dvi’.

HTML can be produced with ‘makeinfo --html’, see section “Generating HTML” in Texinfo. Or alternately ‘texi2html’, see section “About” in Texinfo To HTML.

PDF can be produced with ‘texi2dvi --pdf’ (see section “PDF Output” in Texinfo) or with ‘pdftex’.

Some supplementary notes can be found in the ‘doc’ subdirectory.

2.2 ABI and ISA

ABI (Application Binary Interface) refers to the calling conventions between functions, meaning what registers are used and what sizes the various C data types are. ISA (Instruction Set Architecture) refers to the instructions and registers a CPU has available.

Some 64-bit ISA CPUs have both a 64-bit ABI and a 32-bit ABI defined, the latter for compatibility with older CPUs in the family. GMP supports some CPUs like this in both ABIs. In fact within GMP ‘ABI’ means a combination of chip ABI, plus how GMP chooses to use it. For example in some 32-bit ABIs, GMP may support a limb as either a 32-bit long or a 64-bit long

long.

By default GMP chooses the best ABI available for a given system, and this generally gives significantly greater speed. But an ABI can be chosen explicitly to make GMP compatible with other libraries, or particular application requirements. For example,

     ./configure ABI=32

In all cases it’s vital that all object code used in a given program is compiled for the same ABI.

Usually a limb is implemented as a long. When a long long limb is used this is encoded in the generated ‘gmp.h’. This is convenient for applications, but it does mean that ‘gmp.h’ will vary, and can’t be just copied around. ‘gmp.h’ remains compiler independent though, since all compilers for a particular ABI will be expected to use the same limb type.

Currently no attempt is made to follow whatever conventions a system has for installing library or header files built for a particular ABI. This will probably only matter when installing multiple builds of GMP, and it might be as simple as configuring with a special ‘libdir’, or it might require more than that. Note that builds for different ABIs need to done separately, with a fresh

./configure and make each.

HPPA 2.0 (‘hppa2.0*’)

          ‘ABI=2.0w’

The 2.0w ABI uses 64-bit limbs and pointers and is available on HP-UX 11 or up when using cc. gcc support for this is in progress. Applications must be compiled with

                         cc  +DD64
          ‘ABI=2.0n’

The 2.0n ABI means the 32-bit HPPA 1.0 ABI but with a 64-bit limb using long long. This is available on HP-UX 10 or up when using cc. No gcc support is planned for this. Applications must be compiled with

                         cc  +DA2.0 +e
          ‘ABI=1.0’

HPPA 2.0 CPUs can run all HPPA 1.0 and 1.1 code in the 32-bit HPPA 1.0 ABI. No special compiler options are needed for applications.

All three ABIs are available for CPUs ‘hppa2.0w’ and ‘hppa2.0’, but for CPU ‘hppa2.0n’ only 2.0n or 1.0 are allowed.

MIPS under IRIX 6 (‘mips*-*-irix[6789]’)

IRIX 6 supports the n32 and 64 ABIs and always has a 64-bit MIPS 3 or better CPU. In both these ABIs GMP uses a 64-bit limb. A new enough gcc is required (2.95 for instance).

‘ABI=n32’

The n32 ABI is 32-bit pointers and integers, but with a 64-bit limb using a long long. Applications must be compiled with

     gcc  -mabi=n32
     cc   -n32
‘ABI=64’

The 64-bit ABI is 64-bit pointers and integers. Applications must be compiled with

     gcc  -mabi=64
     cc   -64

Note that MIPS GNU/Linux, as of kernel version 2.2, doesn’t have the necessary support for n32 or 64 and so only gets a 32-bit limb and the MIPS 2 code.

PowerPC 64 (‘powerpc64*’)

‘ABI=aix64’

The AIX 64 ABI uses 64-bit limbs and pointers and is available on systems ‘powerpc64*-*-aix*’. Applications must be compiled (and linked) with

     gcc  -maix64
     xlc  -q64
‘ABI=32L’

This uses the 32-bit ABI but a 64-bit limb using GCC long long in 64-bit registers. Applications must be compiled with

     gcc  -mpowerpc64
‘ABI=32’

This is the basic 32-bit PowerPC ABI. No special compiler options are needed for applications.

Sparc V9 (‘sparcv9’ and ‘ultrasparc*’)
‘ABI=64’

The 64-bit V9 ABI is available on Solaris 2.7 and up and GNU/Linux. GCC 2.95 or up, or Sun cc is required. Applications must be compiled with

     gcc  -m64 -mptr64 -Wa,-xarch=v9 -mcpu=v9
     cc   -xarch=v9
‘ABI=32’

On Solaris 2.6 and earlier, and on Solaris 2.7 with the kernel in 32bit mode, only the plain V8 32-bit ABI can be used, since the kernel doesn’t save all registers. GMP still uses as much of the V9 ISA as it can in these circumstances. No special compiler options are required for applications, though using something like the following requesting V9 code within the V8 ABI is recommended.

     gcc  -mv8plus
     cc   -xarch=v8plus

gcc 2.8 and earlier only supports ‘-mv8’ though.

Don’t be confused by the names of these sparc ‘-m’ and ‘-x’ options, they’re called ‘arch’ but they effectively control the ABI. On Solaris 2.7 with the kernel in 32-bit-mode, a normal native build will reject ‘ABI=64’ because the resulting executables won’t run. ‘ABI=64’ can still be built if desired by making it look like a cross-compile, for example

               ./configure --build=none --host=sparcv9-sun-solaris2.7 ABI=64

2.3 Notes for Package Builds

GMP should present no great difficulties for packaging in a binary distribution.

Libtool is used to build the library and ‘-version-info’ is set appropriately, having started from ‘3:0:0’ in GMP 3.0. The GMP 4 series will be upwardly binary compatible in each release and will be upwardly binary compatible with all of the GMP 3 series. Additional function interfaces may be added in each release, so on systems where libtool versioning is not fully checked by the loader an auxiliary mechanism may be needed to express that a dynamic linked application depends on a new enough GMP.

An auxiliary mechanism may also be needed to express that ‘libgmpxx.la’ (from ‘--enable-cxx’, see Section 2.1 [Build Options], page 3) requires ‘libgmp.la’ from the same GMP version, since this is not done by the libtool versioning, nor otherwise. A mismatch will result in unresolved symbols from the linker, or perhaps the loader.

Using ‘DESTDIR’ or a ‘prefix’ override with ‘make install’ and a shared ‘libgmpxx’ may run into a libtool relinking problem, see Section 2.5 [Known Build Problems], page 12.

When building a package for a CPU family, care should be taken to use ‘--host’ (or ‘--build’) to choose the least common denominator among the CPUs which might use the package. For example this might necessitate ‘i386’ for x86s, or plain ‘sparc’ (meaning V7) for SPARCs.

Users who care about speed will want GMP built for their exact CPU type, to make use of the available optimizations. Providing a way to suitably rebuild a package may be useful. This could be as simple as making it possible for a user to omit ‘--build’ (and ‘--host’) so ‘./config.guess’ will detect the CPU. But a way to manually specify a ‘--build’ will be wanted for systems where ‘./config.guess’ is inexact.

Note that ‘gmp.h’ is a generated file, and will be architecture and ABI dependent.

2.4 Notes for Particular Systems

AIX 3 and 4 On systems ‘*-*-aix[34]*’ shared libraries are disabled by default, since some versions of the native ar fail on the convenience libraries used. A shared build can be attempted with

               ./configure --enable-shared --disable-static

Note that the ‘--disable-static’ is necessary because in a shared build libtool makes ‘libgmp.a’ a symlink to ‘libgmp.so’, apparently for the benefit of old versions of ld which only recognise ‘.a’, but unfortunately this is done even if a fully functional ld is available.

ARM On systems ‘arm*-*-*’, versions of GCC up to and including 2.95.3 have a bug in unsigned division, giving wrong results for some operands. GMP ‘./configure’ will demand GCC 2.95.4 or later.

Microsoft Windows On systems ‘*-*-cygwin*’, ‘*-*-mingw*’ and ‘*-*-pw32*’ by default GMP builds only a static library, but a DLL can be built instead using

               ./configure --disable-static --enable-shared

Static and DLL libraries can’t both be built, since certain export directives in ‘gmp.h’ must be different. ‘--enable-cxx’ cannot be used when building a DLL, since libtool doesn’t currently support C++ DLLs. This might change in the future.

GCC is recommended for compiling GMP, but the resulting DLL can be used with any compiler. On mingw only the standard Windows libraries will be needed, on Cygwin the usual cygwin runtime will be required.

Motorola 68k CPU Types ‘m68k’ is taken to mean 68000. ‘m68020’ or higher will give a performance boost on applicable CPUs. ‘m68360’ can be used for CPU32 series chips. ‘m68302’ can be used for “Dragonball” series chips, though this is merely a synonym for ‘m68000’.

OpenBSD 2.6 m4 in this release of OpenBSD has a bug in eval that makes it unsuitable for ‘.asm’ file processing. ‘./configure’ will detect the problem and either abort or choose another m4 in the PATH. The bug is fixed in OpenBSD 2.7, so either upgrade or use GNU m4.

Power CPU Types In GMP, CPU types ‘power*’ and ‘powerpc*’ will each use instructions not available on the other, so it’s important to choose the right one for the CPU that will be used. Currently GMP has no assembler code support for using just the common instruction subset. To get executables that run on both, the current suggestion is to use the generic C code (CPU ‘none’), possibly with appropriate compiler options (like ‘-mcpu=common’ for gcc). CPU ‘rs6000’ (which is not a CPU but a family of workstations) is accepted by ‘config.sub’, but is currently equivalent to ‘none’.

Sparc CPU Types ‘sparcv8’ or ‘supersparc’ on relevant systems will give a significant performance increase over the V7 code.

SunOS 4 /usr/bin/m4 lacks various features needed to process ‘.asm’ files, and instead ‘./configure’ will automatically use /usr/5bin/m4, which we believe is always available (if not then use GNU m4).

x86 CPU Types ‘i386’ selects generic code which will run reasonably well on all x86 chips. ‘i586’, ‘pentium’ or ‘pentiummmx’ code is good for the intended P5 Pentium chips, but quite slow when run on Intel P6 class chips (PPro, P-II, P-III). ‘i386’ is a better choice when making binaries that must run on both. ‘pentium4’ and an SSE2 capable assembler are important for best results on Pentium 4. The specific code is for instance roughly a 2× to 3× speedup over the generic ‘i386’ code.

x86 MMX and SSE2 Code If the CPU selected has MMX code but the assembler doesn’t support it, a warning is given and non-MMX code is used instead. This will be an inferior build, since the MMX code that’s present is there because it’s faster than the corresponding plain integer code. The same applies to SSE2. Old versions of ‘gas’ don’t support MMX instructions, in particular version 1.92.3 that comes with FreeBSD 2.2.8 doesn’t (and unfortunately there’s no newer assembler for that system). Solaris 2.6 and 2.7 as generate incorrect object code for register to register movq instructions, and so can’t be used for MMX code. Install a recent gas if MMX code is wanted on these systems.

x86 GCC ‘-march=pentiumpro’

GCC 2.95.2 and 2.95.3 miscompiled some versions of ‘mpz/powm.c’ when ‘-march=pentiumpro’ was used, so for relevant CPUs that option is only in the default CFLAGS for GCC 2.95.4 and up.

2.5 Known Build Problems

You might find more up-to-date information at http://swox.com/gmp/.

DJGPP The DJGPP port of bash 2.03 is unable to run the ‘configure’ script, it exits silently, having died writing a preamble to ‘config.log’. Use bash 2.04 or higher. ‘make all’ was found to run out of memory during the final ‘libgmp.la’ link on one system tested, despite having 64Mb available. A separate ‘make libgmp.la’ helped, perhaps recursing into the various subdirectories uses up memory.

‘DESTDIR’ and shared ‘libgmpxx’
‘make install DESTDIR=/my/staging/area’ or the same with a ‘prefix’ override

to install to a temporary directory is not fully supported by current versions of libtool when building a shared version of a library which depends on another being built at the same time, like ‘libgmpxx’ and ‘libgmp’. The problem is that ‘libgmpxx’ is relinked at the install stage to ensure that if the system puts a hard-coded path to ‘libgmp’ within ‘libgmpxx’ then that path will be correct. Naturally the linker is directed to look only at the final location, not the staging area, so if ‘libgmp’ is not already in that final location then the link will fail. On systems which don’t hard-code library paths, for instance SVR4 style systems such as GNU/Linux, a workaround is to insert a suitable -L in the relink_command of ‘libgmpxx.la’ after building but before installing. This can be automated with something like

     sed '/^relink_command/s:libgmp.la:-L /my/staging/area libgmp.la:' \
         <libgmpxx.la >libgmpxx.new
     mv libgmpxx.new libgmpxx.la

GNU binutils strip GNU binutils strip should not be used on the static libraries ‘libgmp.a’ and ‘libmp.a’, neither directly nor via ‘make install-strip’. It can be used on the shared libraries ‘libgmp.so’ and ‘libmp.so’ though. Currently (binutils 2.10.0), strip unpacks an archive then operates on the files, but GMP contains multiple object files of the same name (eg. three versions of ‘init.o’), and they overwrite each other, leaving only the one that happens to be last. If stripped static libraries are wanted, the suggested workaround is to build normally, strip the separate object files, and do another ‘make all’ to rebuild. Alternately ‘CFLAGS’ with ‘-g’ omitted can always be used if it’s just debugging which is unwanted.

make syntax error On certain versions of SCO OpenServer 5 and IRIX 6.5 the native make is unable to handle the long dependencies list for ‘libgmp.la’. The symptom is a “syntax error” on the following line of the top-level ‘Makefile’.

     libgmp.la: $(libgmp_la_OBJECTS) $(libgmp_la_DEPENDENCIES)

Either use GNU Make, or as a workaround remove $(libgmp_la_DEPENDENCIES) from that line (which will make the initial build work, but if any recompiling is done ‘libgmp.la’ might not be rebuilt).

NeXT prior to 3.3 The system compiler on old versions of NeXT was a massacred and old GCC, even if it called itself ‘cc’. This compiler cannot be used to build GMP, you need to get a real GCC, and install that. (NeXT may have fixed this in release 3.3 of their system.)

POWER and PowerPC Bugs in GCC 2.7.2 (and 2.6.3) mean it can’t be used to compile GMP on POWER or PowerPC. If you want to use GCC for these machines, get GCC 2.7.2.1 (or later).

Sequent Symmetry Use the GNU assembler instead of the system assembler, since the latter has serious bugs.

Solaris 2.6 The system sed prints an error “Output line too long” when libtool builds ‘libgmp.la’. This doesn’t seem to cause any obvious ill effects, but GNU sed is recommended, to avoid any doubt.

Sparc Solaris 2.7 with gcc 2.95.2 in ABI=32 A shared library build of GMP seems to fail in this combination, it builds but then fails the tests, apparently due to some incorrect data relocations within gmp_

randinit_lc_2exp_size. The exact cause is unknown, ‘--disable-shared’ is rec-

ommended.

Windows DLL test programs When creating a DLL version of ‘libgmp’, libtool creates wrapper scripts like ‘t-mul’ for programs that would normally be ‘t-mul.exe’, in order to setup the right library paths etc. This works fine, but the absence of ‘t-mul.exe’ etc causes make to think they need recompiling every time, which is an annoyance when re-running a ‘make

check’.

3 GMP Basics

Using functions, macros, data types, etc. not documented in this manual is strongly discouraged. If you do so your application is guaranteed to be incompatible with future versions of GMP.

3.1 Headers and Libraries

All declarations needed to use GMP are collected in the include file ‘gmp.h’. It is designed to work with both C and C++ compilers.

     #include <gmp.h>

Note however that prototypes for GMP functions with FILE * parameters are only provided if <stdio.h> is included too.

     #include <stdio.h>
     #include <gmp.h>

Likewise <stdarg.h> (or <varargs.h>) is required for prototypes with va_list parameters, such as gmp_vprintf. And <obstack.h> for prototypes with struct obstack parameters, such

as gmp_obstack_printf, when available.

All programs using GMP must link against the ‘libgmp’ library. On a typical Unix-like system this can be done with ‘-lgmp’, for example

     gcc myprogram.c -lgmp

GMP C++ functions are in a separate ‘libgmpxx’ library. This is built and installed if C++ support has been enabled (see Section 2.1 [Build Options], page 3). For example,

     g++ mycxxprog.cc -lgmpxx -lgmp

GMP is built using Libtool and an application can use that to link if desired, see section “Introduction” in GNU Libtool

If GMP has been installed to a non-standard location then it may be necessary to use ‘-I’ and ‘-L’ compiler options to point to the right directories, and some sort of run-time path for a shared library. Consult your compiler documentation, for instance section “Introduction” in Using and Porting the GNU Compiler Collection.

3.2 Nomenclature and Types

In this manual, integer usually means a multiple precision integer, as defined by the GMP library. The C data type for such integers is mpz_t. Here are some examples of how to declare such integers:

     mpz_t sum;
     struct foo { mpz_t x, y; };
     mpz_t vec[20];

Rational number means a multiple precision fraction. The C data type for these fractions is mpq_t. For example:

     mpq_t quotient;

Floating point number or Float for short, is an arbitrary precision mantissa with a limited precision exponent. The C data type for such objects is mpf_t.

A limb means the part of a multi-precision number that fits in a single machine word. (We chose this word because a limb of the human body is analogous to a digit, only larger, and containing several digits.) Normally a limb is 32 or 64 bits. The C data type for a limb is mp_limb_t.

3.3 Function Classes

There are six classes of functions in the GMP library:

1. Functions for signed integer arithmetic, with names beginning with mpz_. The associated type is mpz_t. There are about 150 functions in this class.

2. Functions for rational number arithmetic, with names beginning with mpq_. The associated type is mpq_t. There are about 40 functions in this class, but the integer functions can be used for arithmetic on the numerator and denominator separately.

3. Functions for floating-point arithmetic, with names beginning with mpf_. The associated type is mpf_t. There are about 60 functions is this class.

4. Functions compatible with Berkeley MP, such as itom, madd, and mult. The associated type is MINT.

5. Fast low-level functions that operate on natural numbers. These are used by the functions in the preceding groups, and you can also call them directly from very time-critical user programs. These functions’ names begin with mpn_. The associated type is array of mp_ limb_t. There are about 30 (hard-to-use) functions in this class.

6. Miscellaneous functions. Functions for setting up custom allocation and functions for generating random numbers.

3.4 Variable Conventions

GMP functions generally have output arguments before input arguments. This notation is by analogy with the assignment operator. The BSD MP compatibility functions are exceptions, having the output arguments last.

GMP lets you use the same variable for both input and output in one call. For example, the main function for integer multiplication, mpz_mul, can be used to square x and put the result back in x with

     mpz_mul (x, x, x);

Before you can assign to a GMP variable, you need to initialize it by calling one of the special initialization functions. When you’re done with a variable, you need to clear it out, using one of the functions for that purpose. Which function to use depends on the type of variable. See the chapters on integer functions, rational number functions, and floating-point functions for details.

A variable should only be initialized once, or at least cleared between each initialization. After a variable has been initialized, it may be assigned to any number of times.

For efficiency reasons, avoid excessive initializing and clearing. In general, initialize near the start of a function and clear near the end. For example,

     void
     foo (void)
     {
       mpz_t  n;
       int    i;
       mpz_init (n);
       for (i = 1; i < 100; i++)
         {

           mpz_mul (n, ...);
           mpz_fdiv_q (n, ...);
           ...
         }
       mpz_clear (n);
     }

3.5 Parameter Conventions

When a GMP variable is used as a function parameter, it’s effectively a call-by-reference, meaning if the function stores a value there it will change the original in the caller. Parameters which are input-only can be designated const to provoke a compiler error or warning on attempting to modify them.

When a function is going to return a GMP result, it should designate a parameter that it sets, like the library functions do. More than one value can be returned by having more than one output parameter, again like the library functions. A return of an mpz_t etc doesn’t return the object, only a pointer, and this is almost certainly not what’s wanted.

Here’s an example accepting an mpz_t parameter, doing a calculation, and storing the result to the indicated parameter.

     void
     foo (mpz_t result, const mpz_t param, unsigned long n)
     {
       unsigned long  i;
       mpz_mul_ui (result, param, n);
       for (i = 1; i < n; i++)
         mpz_add_ui (result, result, i*7);
     }
     int
     main (void)
     {
       mpz_t  r, n;
       mpz_init (r);
       mpz_init_set_str (n, "123456", 0);
       foo (r, n, 20L);
       gmp_printf ("%Zd\n", r);
       return 0;
     }

foo works even if the mainline passes the same variable for param and result, just like the library functions. But sometimes it’s tricky to make that work, and an application might not want to bother supporting that sort of thing.

For interest, the GMP types mpz_t etc are implemented as one-element arrays of certain structures. This is why declaring a variable creates an object with the fields GMP needs, but then using it as a parameter passes a pointer to the object. Note that the actual fields in each mpz_t etc are for internal use only and should not be accessed directly by code that expects to be compatible with future GMP releases.

3.6 Memory Management

The GMP types like mpz_t are small, containing only a couple of sizes, and pointers to allocated data. Once a variable is initialized, GMP takes care of all space allocation. Additional space is allocated whenever a variable doesn’t have enough.

mpz_t and mpq_t variables never reduce their allocated space. Normally this is the best policy, since it avoids frequent reallocation. Applications that need to return memory to the heap at some particular point can use mpz_realloc2, or clear variables no longer needed.

mpf_t variables, in the current implementation, use a fixed amount of space, determined by the chosen precision and allocated at initialization, so their size doesn’t change.

All memory is allocated using malloc and friends by default, but this can be changed, see Chapter 14 [Custom Allocation], page 78. Temporary memory on the stack is also used (via alloca), but this can be changed at build-time if desired, see Section 2.1 [Build Options], page 3.

3.7 Reentrancy

GMP is reentrant and thread-safe, with some exceptions:

  • If  configured  with  ‘--enable-alloca=malloc-notreentrant’  (or  with

‘--enable-alloca=notreentrant’ when alloca is not available), then naturally GMP is not reentrant.

• mpf_set_default_prec and mpf_init use a global variable for the selected precision. mpf_ init2 can be used instead.

• mpz_random and the other old random number functions use a global random state and are hence not reentrant. The newer random number functions that accept a gmp_randstate_t parameter can be used instead.

• mp_set_memory_functions uses global variables to store the selected memory allocation functions.

• If the memory allocation functions set by a call to mp_set_memory_functions (or malloc and friends by default) are not reentrant, then GMP will not be reentrant either.

• If the standard I/O functions such as fwrite are not reentrant then the GMP I/O functions using them will not be reentrant either.

• It’s safe for two threads to read from the same GMP variable simultaneously, but it’s not safe for one to read while the another might be writing, nor for two threads to write simultaneously. It’s not safe for two threads to generate a random number from the same gmp_randstate_t simultaneously, since this involves an update of that variable.

• On SCO systems the default <ctype.h> macros use per-file static variables and may not be reentrant, depending whether the compiler optimizes away fetches from them. The GMP text-based input functions are affected.

3.8 Useful Macros and Constants

Global Constant const int mp bits per limb

The number of bits per limb.

Macro GNU MP VERSION

Macro GNU MP VERSION MINOR

Macro GNU MP VERSION PATCHLEVEL

The major and minor GMP version, and patch level, respectively, as integers. For GMP i.j, these numbers will be i, j, and 0, respectively. For GMP i.j.k, these numbers will be i, j, and k, respectively.

Global Constant const char * const gmp version

The GMP version number, as a null-terminated string, in the form “i.j” or “i.j.k”. This release is "4.1".

3.9 Compatibility with older versions

This version of GMP is upwardly binary compatible with all 4.x and 3.x versions, and upwardly compatible at the source level with all 2.x versions, with the following exceptions.

• mpn_gcd had its source arguments swapped as of GMP 3.0, for consistency with other mpn functions.

• mpf_get_prec counted precision slightly differently in GMP 3.0 and 3.0.1, but in 3.1 reverted to the 2.x style.

There are a number of compatibility issues between GMP 1 and GMP 2 that of course also apply when porting applications from GMP 1 to GMP 4. Please see the GMP 2 manual for details.

The Berkeley MP compatibility library (see Chapter 13 [BSD Compatible Functions], page 76) is source and binary compatible with the standard ‘libmp’.

3.10 Efficiency

Small operands On small operands, the time for function call overheads and memory allocation can be significant in comparison to actual calculation. This is unavoidable in a general purpose variable precision library, although GMP attempts to be as efficient as it can on both large and small operands.

Static Linking On some CPUs, in particular the x86s, the static ‘libgmp.a’ should be used for maximum speed, since the PIC code in the shared ‘libgmp.so’ will have a small overhead on each function call and global data address. For many programs this will be insignificant, but for long calculations there’s a gain to be had.

Initializing and clearing Avoid excessive initializing and clearing of variables, since this can be quite time consuming, especially in comparison to otherwise fast operations like addition. A language interpreter might want to keep a free list or stack of initialized variables ready for use. It should be possible to integrate something like that with a garbage collector too.

Reallocations An mpz_t or mpq_t variable used to hold successively increasing values will have its memory repeatedly realloced, which could be quite slow or could fragment memory, depending on the C library. If an application can estimate the final size then mpz_init2 or mpz_realloc2 can be called to allocate the necessary space from the beginning (see Section 5.1 [Initializing Integers], page 25). It doesn’t matter if a size set with mpz_init2 or mpz_realloc2 is too small, since all functions will do a further reallocation if necessary. Badly overestimating memory required will waste space though.

2exp functions It’s up to an application to call functions like mpz_mul_2exp when appropriate. General purpose functions like mpz_mul make no attempt to identify powers of two or other special forms, because such inputs will usually be very rare and testing every time would be wasteful. ui and si functions

The ui functions and the small number of si functions exist for convenience and should be used where applicable. But if for example an mpz_t contains a value that fits in an unsigned long there’s no need extract it and call a ui function, just use the regular mpz function.

In-Place Operations

mpz_abs, mpq_abs, mpf_abs, mpz_neg, mpq_neg and mpf_neg are fast when used for

in-place operations like mpz_abs(x,x), since in the current implementation only a single field of x needs changing. On suitable compilers (GCC for instance) this is inlined too.

mpz_add_ui, mpz_sub_ui, mpf_add_ui and mpf_sub_ui benefit from an in-place

operation like mpz_add_ui(x,x,y), since usually only one or two limbs of x will need to be changed. The same applies to the full precision mpz_add etc if y is small. If y is big then cache locality may be helped, but that’s all. mpz_mul is currently the opposite, a separate destination is slightly better. A call like mpz_mul(x,x,y) will, unless y is only one limb, make a temporary copy of x before forming the result. Normally that copying will only be a tiny fraction of the time for the multiply, so this is not a particularly important consideration. mpz_set, mpq_set, mpq_set_num, mpf_set, etc, make no attempt to recognise a copy of something to itself, so a call like mpz_set(x,x) will be wasteful. Naturally that would never be written deliberately, but if it might arise from two pointers to the same object then a test to avoid it might be desirable.

     if (x != y)
       mpz_set (x, y);

Note that it’s never worth introducing extra mpz_set calls just to get in-place operations. If a result should go to a particular variable then just direct it there and let GMP take care of data movement.

Divisibility Testing (Small Integers)

mpz_divisible_ui_p and mpz_congruent_ui_p are the best functions for testing

whether an mpz_t is divisible by an individual small integer. They use an algorithm which is faster than mpz_tdiv_ui, but which gives no useful information about the actual remainder, only whether it’s zero (or a particular value). However when testing divisibility by several small integers, it’s best to take a remainder modulo their product, to save multi-precision operations. For instance to test whether a number is divisible by any of 23, 29 or 31 take a remainder modulo 23 × 29 × 31 = 20677 and then test that. The division functions like mpz_tdiv_q_ui which give a quotient as well as a remainder are generally a little slower than the remainder-only functions like mpz_ tdiv_ui. If the quotient is only rarely wanted then it’s probably best to just take a remainder and then go back and calculate the quotient if and when it’s wanted (mpz_divexact_ui can be used if the remainder is zero).

Rational Arithmetic The mpq functions operate on mpq_t values with no common factors in the numerator and denominator. Common factors are checked-for and cast out as necessary. In general, cancelling factors every time is the best approach since it minimizes the sizes for subsequent operations. However, applications that know something about the factorization of the values they’re working with might be able to avoid some of the GCDs used for canonicalization, or swap them for divisions. For example when multiplying by a prime it’s enough to check for factors of it in the denominator instead of doing a full GCD. Or when forming a big product it might be known that very little cancellation will be possible, and so canonicalization can be left to the end.

The mpq_numref and mpq_denref macros give access to the numerator and denominator to do things outside the scope of the supplied mpq functions. See Section 6.5 [Applying Integer Functions], page 41. The canonical form for rationals allows mixed-type mpq_t and integer additions or subtractions to be done directly with multiples of the denominator. This will be somewhat faster than mpq_add. For example,

     /* mpq increment */
     mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q));
     /* mpq += unsigned long */
     mpz_addmul_ui (mpq_numref(q), mpq_denref(q), 123UL);
     /* mpq -= mpz */
     mpz_submul (mpq_numref(q), mpq_denref(q), z);

Number Sequences Functions like mpz_fac_ui, mpz_fib_ui and mpz_bin_uiui are designed for calculating isolated values. If a range of values is wanted it’s probably best to call to get a starting point and iterate from there.

Text Input/Output Hexadecimal or octal are suggested for input or output in text form. Power-of-2 bases like these can be converted much more efficiently than other bases, like decimal. For big numbers there’s usually nothing of particular interest to be seen in the digits, so the base doesn’t matter much. Maybe we can hope octal will one day become the normal base for everyday use, as proposed by King Charles XII of Sweden and later reformers.

3.11 Debugging

Stack Overflow Depending on the system, a segmentation violation or bus error might be the only indication of stack overflow. See ‘--enable-alloca’ choices in Section 2.1 [Build Options], page 3, for how to address this.

Heap Problems The most likely cause of application problems with GMP is heap corruption. Failing to init GMP variables will have unpredictable effects, and corruption arising elsewhere in a program may well affect GMP. Initializing GMP variables more than once or failing to clear them will cause memory leaks. In all such cases a malloc debugger is recommended. On a GNU or BSD system the standard C library malloc has some diagnostic facilities, see section “Allocation Debugging” in The GNU C Library Reference Manual, or ‘man 3 malloc’. Other possibilities, in no particular order, include

     http://www.inf.ethz.ch/personal/biere/projects/ccmalloc
     http://quorum.tamu.edu/jon/gnu (debauch)
     http://dmalloc.com
     http://www.perens.com/FreeSoftware (electric fence)
     http://packages.debian.org/fda
     http://www.gnupdate.org/components/leakbug
     http://people.redhat.com/~otaylor/memprof
     http://www.cbmamiga.demon.co.uk/mpatrol

The GMP default allocation routines in ‘memory.c’ also have a simple sentinel scheme which can be enabled with #define DEBUG in that file. This is mainly designed for detecting buffer overruns during GMP development, but might find other uses.

Stack Backtraces On some systems the compiler options GMP uses by default can interfere with debugging. In particular on x86 and 68k systems ‘-fomit-frame-pointer’ is used and this generally inhibits stack backtracing. Recompiling without such options may help while debugging, though the usual caveats about it potentially moving a memory problem or hiding a compiler bug will apply.

GNU Debugger A sample ‘.gdbinit’ is included in the distribution, showing how to call some undocumented dump functions to print GMP variables from within GDB. Note that these functions shouldn’t be used in final application code since they’re undocumented and may be subject to incompatible changes in future versions of GMP.

Source File Paths GMP has multiple source files with the same name, in different directories. For example ‘mpz’, ‘mpq’, ‘mpf’ and ‘mpfr’ each have an ‘init.c’. If the debugger can’t already determine the right one it may help to build with absolute paths on each C file. One way to do that is to use a separate object directory with an absolute path to the source directory.

     cd /my/build/dir
     /my/source/dir/gmp-4.1/configure

This works via VPATH, and might require GNU make. Alternately it might be possible to change the .c.lo rules appropriately.

Assertion Checking The build option ‘--enable-assert’ is available to add some consistency checks to the library (see Section 2.1 [Build Options], page 3). These are likely to be of limited value to most applications. Assertion failures are just as likely to indicate memory corruption as a library or compiler bug. Applications using the low-level mpn functions, however, will benefit from ‘--enable-assert’ since it adds checks on the parameters of most such functions, many of which have subtle restrictions on their usage. Note however that only the generic C code has checks, not the assembler code, so CPU ‘none’ should be used for maximum checking.

Temporary Memory Checking The build option ‘--enable-alloca=debug’ arranges that each block of temporary memory in GMP is allocated with a separate call to malloc (or the allocation

function set with mp_set_memory_functions).

This can help a malloc debugger detect accesses outside the intended bounds, or detect memory not released. In a normal build, on the other hand, temporary memory is allocated in blocks which GMP divides up for its own use, or may be allocated with a compiler builtin alloca which will go nowhere near any malloc debugger hooks.

Checker The checker program (http://savannah.gnu.org/projects/checker) can be used

with GMP. It contains a stub library which means GMP applications compiled with checker can use a normal GMP build. A build of GMP with checking within GMP itself can be made. This will run very very slowly. Configure with

     ./configure --host=none-pc-linux-gnu CC=checkergcc

‘--host=none’ must be used, since the GMP assembler code doesn’t support the checking scheme. The GMP C++ features cannot be used, since current versions of checker (0.9.9.1) don’t yet support the standard C++ library.

Valgrind The valgrind program (http://devel-home.kde.org/~sewardj) is a memory checker for x86s. It translates and emulates machine instructions to do strong checks for uninitialized data (at the level of individual bits), memory accesses through bad pointers, and memory leaks. Current versions (20020226 snapshot) don’t support MMX or SSE, so GMP must be configured for an x86 without those (eg. plain ‘i386’), or with a special MPN_PATH that excludes those subdirectories (see Section 2.1 [Build Options], page 3).

Other Problems Any suspected bug in GMP itself should be isolated to make sure it’s not an application problem, see Chapter 4 [Reporting Bugs], page 24.

3.12 Profiling

Running a program under a profiler is a good way to find where it’s spending most time and where improvements can be best sought.

Depending on the system, it may be possible to get a flat profile, meaning simple timer sampling of the program counter, with no special GMP build options, just a ‘-p’ when compiling the mainline. This is a good way to ensure minimum interference with normal operation. The necessary symbol type and size information exists in most of the GMP assembler code.

The ‘--enable-profiling’ build option can be used to add suitable compiler flags, either for prof (‘-p’) or gprof (‘-pg’), see Section 2.1 [Build Options], page 3. Which of the two is available and what they do will depend on the system, and possibly on support available in ‘libc’. For some systems appropriate corresponding mcount calls are added to the assembler code too.

On x86 systems prof gives call counting, so that average time spent in a function can be determined. gprof, where supported, adds call graph construction, so for instance calls to mpn_add_n from mpz_add and from mpz_mul can be differentiated.

On x86 and 68k systems ‘-pg’ and ‘-fomit-frame-pointer’ are incompatible, so the latter is not used when gprof profiling is selected, which may result in poorer code generation. If prof profiling is selected instead it should still be possible to use gprof, but only the ‘gprof -p’ flat profile and call counts can be expected to be valid, not the ‘gprof -q’ call graph.

3.13 Autoconf

Autoconf based applications can easily check whether GMP is installed. The only thing to be noted is that GMP library symbols from version 3 onwards have prefixes like __gmpz. The following therefore would be a simple test,

     AC_CHECK_LIB(gmp, __gmpz_init)

This just uses the default AC_CHECK_LIB actions for found or not found, but an application that must have GMP would want to generate an error if not found. For example,

     AC_CHECK_LIB(gmp, __gmpz_init, , [AC_MSG_ERROR(
     [GNU MP not found, see http://swox.com/gmp])])

If functions added in some particular version of GMP are required, then one of those can be used when checking. For example mpz_mul_si was added in GMP 3.1,

     AC_CHECK_LIB(gmp, __gmpz_mul_si, , [AC_MSG_ERROR(
     [GNU MP not found, or not 3.1 or up, see http://swox.com/gmp])])

An alternative would be to test the version number in ‘gmp.h’ using say AC_EGREP_CPP. That would make it possible to test the exact version, if some particular sub-minor release is known to be necessary.

An application that can use either GMP 2 or 3 will need to test for __gmpz_init (GMP 3 and up) or mpz_init (GMP 2), and it’s also worth checking for ‘libgmp2’ since Debian GNU/Linux systems used that name in the past. For example,

     AC_CHECK_LIB(gmp, __gmpz_init, ,
       [AC_CHECK_LIB(gmp, mpz_init, ,
         [AC_CHECK_LIB(gmp2, mpz_init)])])

In general it’s suggested that applications should simply demand a new enough GMP rather than trying to provide supplements for features not available in past versions.

Occasionally an application will need or want to know the size of a type at configuration or preprocessing time, not just with sizeof in the code. This can be done in the normal way with mp_limb_t etc, but GMP 4.0 or up is best for this, since prior versions needed certain ‘-D’ defines on systems using a long long limb. The following would suit Autoconf 2.50 or up,

     AC_CHECK_SIZEOF(mp_limb_t, , [#include <gmp.h>])

The optional mpfr functions are provided in a separate ‘libmpfr.a’, and this might be from GMP with ‘--enable-mpfr’ or from MPFR installed separately. Either way ‘libmpfr’ depends on ‘libgmp’, it doesn’t stand alone. Currently only a static ‘libmpfr.a’ will be available, not a shared library, since upward binary compatibility is not guaranteed.

     AC_CHECK_LIB(mpfr, mpfr_add, , [AC_MSG_ERROR(
     [Need MPFR either from GNU MP 4 or separate MPFR package.
     See http://www.mpfr.org or http://swox.com/gmp])

4 Reporting Bugs

If you think you have found a bug in the GMP library, please investigate it and report it. We have made this library available to you, and it is not too much to ask you to report the bugs you find.

Before you report a bug, check it’s not already addressed in Section 2.5 [Known Build Problems], page 12, or perhaps Section 2.4 [Notes for Particular Systems], page 10. You may also want to check http://swox.com/gmp/ for patches for this release.

Please include the following in any report,

• The GMP version number, and if pre-packaged or patched then say so.

• A test program that makes it possible for us to reproduce the bug. Include instructions on how to run the program.

• A description of what is wrong. If the results are incorrect, in what way. If you get a crash, say so.

• If you get a crash, include a stack backtrace from the debugger if it’s informative (‘where’ in gdb, or ‘$C’ in adb).

• Please do not send core dumps, executables or straces.

• The configuration options you used when building GMP, if any.

• The name of the compiler and its version. For gcc, get the version with ‘gcc -v’, otherwise perhaps ‘what `which cc`’, or similar.

• The output from running ‘uname -a’.

• The output from running ‘./config.guess’, and from running ‘./configfsf.guess’ (might be the same).

• If the bug is related to ‘configure’, then the contents of ‘config.log’.

• If the bug is related to an ‘asm’ file not assembling, then the contents of ‘config.m4’ and the offending line or lines from the temporary ‘mpn/tmp-<file>.s’.

Please make an effort to produce a self-contained report, with something definite that can be tested or debugged. Vague queries or piecemeal messages are difficult to act on and don’t help the development effort.

It is not uncommon that an observed problem is actually due to a bug in the compiler; the GMP code tends to explore interesting corners in compilers.

If your bug report is good, we will do our best to help you get a corrected version of the library; if the bug report is poor, we won’t do anything about it (except maybe ask you to send a better report).

Send your report to: bug-gmp@gnu.org.

If you think something in this manual is unclear, or downright incorrect, or if the language needs to be improved, please send a note to the same address.

5 Integer Functions

This chapter describes the GMP functions for performing integer arithmetic. These functions start with the prefix mpz_.

GMP integers are stored in objects of type mpz_t.

5.1 Initialization Functions

The functions for integer arithmetic assume that all integer objects are initialized. You do that by calling the function mpz_init. For example,

{
  mpz_t integ;
  mpz_init (integ);
  ...
  mpz_add (integ, ...);
  ...
  mpz_sub (integ, ...);
  /* Unless the program is about to exit, do ... */
  mpz_clear (integ);
}

As you can see, you can store new values any number of times, once an object is initialized.

Function void mpz init (mpz_t integer)

Initialize integer, and set its value to 0.

Function void mpz init2 (mpz_t integer, unsigned long n)

Initialize integer, with space for n bits, and set its value to 0.

n is only the initial space, integer will grow automatically in the normal way, if necessary, for subsequent values stored. mpz_init2 makes it possible to avoid such reallocations if a maximum size is known in advance.

Function void mpz clear (mpz_t integer)

Free the space occupied by integer. Call this function for all mpz_t variables when you are done with them.

Function void mpz realloc2 (mpz_t integer, unsigned long n)

Change the space allocated for integer to n bits. The value in integer is preserved if it fits, or is set to 0 if not.

This function can be used to increase the space for a variable in order to avoid repeated automatic reallocations, or to decrease it to give memory back to the heap.

Function void mpz array init (mpz_t integer array[], size_t array size,

mp_size_t fixed num bits) This is a special type of initialization. Fixed space of fixed num bits bits is allocated to each of the array size integers in integer array.

The space will not be automatically increased, unlike the normal mpz_init, but instead an application must ensure it’s sufficient for any value stored. The following space requirements apply to various functions,

  • mpz_abs, mpz_neg, mpz_set, mpz_set_si and mpz_set_ui need room for the value

they store.

  • mpz_add, mpz_add_ui, mpz_sub and mpz_sub_ui need room for the larger of the two

operands, plus an extra mp_bits_per_limb.

• mpz_mul, mpz_mul_ui and mpz_mul_ui need room for the sum of the number of bits in their operands, but each rounded up to a multiple of mp_bits_per_limb.

• mpz_swap can be used between two array variables, but not between an array and a normal variable.

For other functions, or if in doubt, the suggestion is to calculate in a regular mpz_init variable and copy the result to an array variable with mpz_set.

mpz_array_init can reduce memory usage in algorithms that need large arrays of integers, since it avoids allocating and reallocating lots of small memory blocks. There is no way to free the storage allocated by this function. Don’t call mpz_clear!

Function void * mpz realloc (mpz_t integer, mp_size_t new alloc)

Change the space for integer to new alloc limbs. The value in integer is preserved if it fits, or is set to 0 if not. The return value is not useful to applications and should be ignored.

mpz_realloc2 is the preferred way to accomplish allocation changes like this. mpz_ realloc2 and _mpz_realloc are the same except that _mpz_realloc takes the new size in limbs.

5.2 Assignment Functions

These functions assign new values to already initialized integers (see Section 5.1 [Initializing Integers], page 25).

Function void mpz set (mpz_t rop, mpz_t op)

Function void mpz set ui (mpz_t rop, unsigned long int op)
Function void mpz set si (mpz_t rop, signed long int op)

Function void mpz set d (mpz_t rop, double op) Function void mpz set q (mpz_t rop, mpq_t op) Function void mpz set f (mpz_t rop, mpf_t op)

Set the value of rop from op.

mpz_set_d, mpz_set_q and mpz_set_f truncate op to make it an integer.

Function int mpz set str (mpz_t rop, char *str, int base)

Set the value of rop from str, a null-terminated C string in base base. White space is allowed in the string, and is simply ignored. The base may vary from 2 to 36. If base is 0, the actual base is determined from the leading characters: if the first two characters are “0x” or “0X”, hexadecimal is assumed, otherwise if the first character is “0”, octal is assumed, otherwise decimal is assumed.

This function returns 0 if the entire string is a valid number in base base. Otherwise it returns −1.

[It turns out that it is not entirely true that this function ignores white-space. It does ignore it between digits, but not after a minus sign or within or after “0x”. We are considering changing the definition of this function, making it fail when there is any white-space in the input, since that makes a lot of sense. Send your opinion of this change to bug-gmp@gnu.org. Do you really want it to accept "3 14" as meaning 314 as it does now?] Function void mpz swap (mpz_t rop1, mpz_t rop2)

Swap the values rop1 and rop2 efficiently.

5.3 Combined Initialization and Assignment Functions

For convenience, GMP provides a parallel series of initialize-and-set functions which initialize the output and then store the value there. These functions’ names have the form mpz_init_set...

Here is an example of using one:

     {
       mpz_t pie;
       mpz_init_set_str (pie, "3141592653589793238462643383279502884", 10);
       ...
       mpz_sub (pie, ...);
       ...
       mpz_clear (pie);
     }

Once the integer has been initialized by any of the mpz_init_set... functions, it can be used as the source or destination operand for the ordinary integer functions. Don’t use an initializeand-set function on a variable already initialized!

Function void mpz init set (mpz_t rop, mpz_t op) Function void mpz init set ui (mpz_t rop, unsigned long int op) Function void mpz init set si (mpz_t rop, signed long int op) Function void mpz init set d (mpz_t rop, double op)

Initialize rop with limb space and set the initial numeric value from op.

Function int mpz init set str (mpz_t rop, char *str, int base)

Initialize rop and set its value like mpz_set_str (see its documentation above for details).

If the string is a correct base base number, the function returns 0; if an error occurs it returns −1. rop is initialized even if an error occurs. (I.e., you have to call mpz_clear for it.)

5.4 Conversion Functions

This section describes functions for converting GMP integers to standard C types. Functions for converting to GMP integers are described in Section 5.2 [Assigning Integers], page 26 and Section 5.12 [I/O of Integers], page 35.

Function unsigned long int mpz get ui (mpz_t op)

Return the value of op as an unsigned long.

If op is too big to fit an unsigned long then just the least significant bits that do fit are returned. The sign of op is ignored, only the absolute value is used.

Function signed long int mpz get si (mpz_t op)

If op fits into a signed long int return the value of op. Otherwise return the least significant part of op, with the same sign as op.

If op is too big to fit in a signed long int, the returned result is probably not very useful. To find out if the value will fit, use the function mpz_fits_slong_p.

Function double mpz get d (mpz_t op)

Convert op to a double.

Function double mpz get d 2exp (signed long int *exp, mpz_t op)

Find d and exp such that d × 2^{exp}, with 0.5 ≤ |d| < 1, is a good approximation to op.

Function char * mpz get str (char *str, int base, mpz_t op)

Convert op to a string of digits in base base. The base may vary from 2 to 36.

If str is NULL, the result string is allocated using the current allocation function (see Chapter 14 [Custom Allocation], page 78). The block will be strlen(str)+1 bytes, that being exactly enough for the string and null-terminator.

If str is not NULL, it should point to a block of storage large enough for the result, that being mpz_sizeinbase (op, base) + 2. The two extra bytes are for a possible minus sign, and the null-terminator.

A pointer to the result string is returned, being either the allocated block, or the given str.

Function mp_limb_t mpz getlimbn (mpz_t op, mp_size_t n)

Return limb number n from op. The sign of op is ignored, just the absolute value is used. The least significant limb is number 0.

mpz_size can be used to find how many limbs make up op. mpz_getlimbn returns zero if n is outside the range 0 to mpz_size(op)-1.

5.5 Arithmetic Functions

Function void mpz add (mpz_t rop, mpz_t op1, mpz_t op2)

Function void mpz add ui (mpz_t rop, mpz_t op1, unsigned long int op2)

Set rop to op1 + op2.

Function void mpz sub (mpz_t rop, mpz_t op1, mpz_t op2)

Function void mpz sub ui (mpz_t rop, mpz_t op1, unsigned long int op2)
Function void mpz ui sub (mpz_t rop, unsigned long int op1, mpz_t op2)

Set rop to op1 − op2.

Function void mpz mul (mpz_t rop, mpz_t op1, mpz_t op2) Function void mpz mul si (mpz_t rop, mpz_t op1, long int op2)

Function void mpz mul ui (mpz_t rop, mpz_t op1, unsigned long int op2)

Set rop to op1 × op2.

Function void mpz addmul (mpz_t rop, mpz_t op1, mpz_t op2)

Function void mpz addmul ui (mpz_t rop, mpz_t op1, unsigned long int op2)

Set rop to rop + op1 × op2.

Function void mpz submul (mpz_t rop, mpz_t op1, mpz_t op2)

Function void mpz submul ui (mpz_t rop, mpz_t op1, unsigned long int op2)

Set rop to rop − op1 × op2.

Function void mpz mul 2exp (mpz_t rop, mpz_t op1, unsigned long int op2)

Set rop to op1 × 2^{op2}. This operation can also be defined as a left shift by op2 bits.

Function void mpz neg (mpz_t rop, mpz_t op)

Set rop to −op.

Function void mpz abs (mpz_t rop, mpz_t op)

Set rop to the absolute value of op.

5.6 Division Functions

Division is undefined if the divisor is zero. Passing a zero divisor to the division or modulo functions (including the modular powering functions mpz_powm and mpz_powm_ui), will cause an intentional division by zero. This lets a program handle arithmetic exceptions in these functions the same way as for normal C int arithmetic.

Function void mpz cdiv q (mpz_t q, mpz_t n, mpz_t d)
Function void mpz cdiv r (mpz_t r, mpz_t n, mpz_t d)
Function void mpz cdiv qr (mpz_t q, mpz_t r, mpz_t n, mpz_t d)
Function unsigned long int mpz cdiv q ui (mpz_t q, mpz_t n,
         unsigned long int d)
Function unsigned long int mpz cdiv r ui (mpz_t r, mpz_t n,
         unsigned long int d)
Function unsigned long int mpz cdiv qr ui (mpz_t q, mpz_t r, mpz_t n,
         unsigned long int d)
Function unsigned long int mpz cdiv ui (mpz_t n, unsigned long int d)
Function void mpz cdiv q 2exp (mpz_t q, mpz_t n, unsigned long int b)
Function void mpz cdiv r 2exp (mpz_t r, mpz_t n, unsigned long int b)
Function void mpz fdiv q (mpz_t q, mpz_t n, mpz_t d)
Function void mpz fdiv r (mpz_t r, mpz_t n, mpz_t d)
Function void mpz fdiv qr (mpz_t q, mpz_t r, mpz_t n, mpz_t d)
Function unsigned long int mpz fdiv q ui (mpz_t q, mpz_t n,
         unsigned long int d)
Function unsigned long int mpz fdiv r ui (mpz_t r, mpz_t n,
         unsigned long int d)
Function unsigned long int mpz fdiv qr ui (mpz_t q, mpz_t r, mpz_t n,
         unsigned long int d)
Function unsigned long int mpz fdiv ui (mpz_t n, unsigned long int d)
Function void mpz fdiv q 2exp (mpz_t q, mpz_t n, unsigned long int b)
Function void mpz fdiv r 2exp (mpz_t r, mpz_t n, unsigned long int b)
Function void mpz tdiv q (mpz_t q, mpz_t n, mpz_t d)
Function void mpz tdiv r (mpz_t r, mpz_t n, mpz_t d)
Function void mpz tdiv qr (mpz_t q, mpz_t r, mpz_t n, mpz_t d)
Function unsigned long int mpz tdiv q ui (mpz_t q, mpz_t n,
         unsigned long int d)
Function unsigned long int mpz tdiv r ui (mpz_t r, mpz_t n,
         unsigned long int d)
Function unsigned long int mpz tdiv qr ui (mpz_t q, mpz_t r, mpz_t n,
         unsigned long int d)
Function unsigned long int mpz tdiv ui (mpz_t n, unsigned long int d)

Function void mpz tdiv q 2exp (mpz_t q, mpz_t n, unsigned long int b)
Function void mpz tdiv r 2exp (mpz_t r, mpz_t n, unsigned long int b)

Divide n by d, forming a quotient q and/or remainder r. For the 2exp functions, d = 2^{b}. The rounding is in three styles, each suiting different applications.

• cdiv rounds q up towards +∞, and r will have the opposite sign to d. The c stands for “ceil”.

• fdiv rounds q down towards −∞, and r will have the same sign as d. The f stands for “floor”.

• tdiv rounds q towards zero, and r will have the same sign as n. The t stands for “truncate”.

In all cases q and r will satisfy n = qd + r, and r will satisfy 0 ≤ |r| < |d|.

The q functions calculate only the quotient, the r functions only the remainder, and the qr functions calculate both. Note that for qr the same variable cannot be passed for both q and r, or results will be unpredictable.

For the ui variants the return value is the remainder, and in fact returning the remainder is all the div_ui functions do. For tdiv and cdiv the remainder can be negative, so for those the return value is the absolute value of the remainder.

The 2exp functions are right shifts and bit masks, but of course rounding the same as the other functions. For positive n both mpz_fdiv_q_2exp and mpz_tdiv_q_2exp are simple bitwise right shifts. For negative n, mpz_fdiv_q_2exp is effectively an arithmetic right shift treating n as twos complement the same as the bitwise logical functions do, whereas mpz_tdiv_q_2exp effectively treats n as sign and magnitude.

Function void mpz mod (mpz_t r, mpz_t n, mpz_t d)
Function unsigned long int mpz mod ui (mpz_t r, mpz_t n,
    unsigned long int d)

Set r to n mod d. The sign of the divisor is ignored; the result is always non-negative.

mpz_mod_ui is identical to mpz_fdiv_r_ui above, returning the remainder as well as setting r. See mpz_fdiv_ui above if only the return value is wanted.

Function void mpz divexact (mpz_t q, mpz_t n, mpz_t d)

Function void mpz divexact ui (mpz_t q, mpz_t n, unsigned long d)

Set q to n/d. These functions produce correct results only when it is known in advance that d divides n.

These routines are much faster than the other division functions, and are the best choice when exact division is known to occur, for example reducing a rational to lowest terms.

Function int mpz divisible p (mpz_t n, mpz_t d)

Function int mpz divisible ui p (mpz_t n, unsigned long int d) Function int mpz divisible 2exp p (mpz_t n, unsigned long int b)

Return non-zero if n is exactly divisible by d, or in the case of mpz_divisible_2exp_p by 2^{b}.

Function int mpz congruent p (mpz_t n, mpz_t c, mpz_t d)

Function int mpz congruent ui p (mpz_t n, unsigned long int c, unsigned
    long int d)
Function int mpz congruent 2exp p (mpz_t n, mpz_t c, unsigned long int b)

Return non-zero if n is congruent to c modulo d, or in the case of mpz_congruent_2exp_p modulo 2^{b}.

5.7 Exponentiation Functions

Function void mpz powm (mpz_t rop, mpz_t base, mpz_t exp, mpz_t mod)

Function void mpz powm ui (mpz_t rop, mpz_t base, unsigned long int exp,
    mpz_t mod)

Set rop to base^{exp} mod mod.

Negative exp is supported if an inverse base^{−1} mod mod exists (see mpz_invert in Section 5.9 [Number Theoretic Functions], page 32). If an inverse doesn’t exist then a divide by zero is raised.

Function void mpz pow ui (mpz_t rop, mpz_t base, unsigned long int exp)
Function void mpz ui pow ui (mpz_t rop, unsigned long int base, unsigned
    long int exp)

Set rop to base^{exp}. The case 0^{0} yields 1.

5.8 Root Extraction Functions

Function int mpz root (mpz_t rop, mpz_t op, unsigned long int n)

Set rop to ⌊ ^{n} \sqrt{op}⌋, the truncated integer part of the nth root of op. Return non-zero if the computation was exact, i.e., if op is rop to the nth power.

Function void mpz sqrt (mpz_t rop, mpz_t op)

Set rop to ⌊\sqrt{op}⌋, the truncated integer part of the square root of op.

Function void mpz sqrtrem (mpz_t rop1, mpz_t rop2, mpz_t op)

Set rop1 to ⌊\sqrt{op}⌋, like mpz_sqrt. Set rop2 to the remainder (op − rop1^{2}), which will be zero if op is a perfect square.

If rop1 and rop2 are the same variable, the results are undefined.

Function int mpz perfect power p (mpz_t op)

Return non-zero if op is a perfect power, i.e., if there exist integers a and b, with b > 1, such that op = a^{b}.

Under this definition both 0 and 1 are considered to be perfect powers. Negative values of op are accepted, but of course can only be odd perfect powers.

Function int mpz perfect square p (mpz_t op)

Return non-zero if op is a perfect square, i.e., if the square root of op is an integer. Under this definition both 0 and 1 are considered to be perfect squares.

5.9 Number Theoretic Functions

Function int mpz probab prime p (mpz_t n, int reps)

Determine whether n is prime. Return 2 if n is definitely prime, return 1 if n is probably prime (without being certain), or return 0 if n is definitely composite.

This function does some trial divisions, then some Miller-Rabin probabilistic primality tests. reps controls how many such tests are done, 5 to 10 is a reasonable number, more will reduce the chances of a composite being returned as “probably prime”.

Miller-Rabin and similar tests can be more properly called compositeness tests. Numbers which fail are known to be composite but those which pass might be prime or might be composite. Only a few composites pass, hence those which pass are considered probably prime.

Function void mpz nextprime (mpz_t rop, mpz_t op)

Set rop to the next prime greater than op.

This function uses a probabilistic algorithm to identify primes. For practical purposes it’s adequate, the chance of a composite passing will be extremely small.

Function void mpz gcd (mpz_t rop, mpz_t op1, mpz_t op2)

Set rop to the greatest common divisor of op1 and op2. The result is always positive even if one or both input operands are negative.

Function unsigned long int mpz gcd ui (mpz_t rop, mpz_t op1, unsigned long

int op2) Compute the greatest common divisor of op1 and op2. If rop is not NULL, store the result there.

If the result is small enough to fit in an unsigned long int, it is returned. If the result does not fit, 0 is returned, and the result is equal to the argument op1. Note that the result will always fit if op2 is non-zero.

Function void mpz gcdext (mpz_t g, mpz_t s, mpz_t t, mpz_t a, mpz_t b)

Set g to the greatest common divisor of a and b, and in addition set s and t to coefficients satisfying as + bt = g. g is always positive, even if one or both of a and b are negative.

If t is NULL then that value is not computed.

Function void mpz lcm (mpz_t rop, mpz_t op1, mpz_t op2)

Function void mpz lcm ui (mpz_t rop, mpz_t op1, unsigned long op2)

Set rop to the least common multiple of op1 and op2. rop is always positive, irrespective of the signs of op1 and op2. rop will be zero if either op1 or op2 is zero.

Function int mpz invert (mpz_t rop, mpz_t op1, mpz_t op2)

Compute the inverse of op1 modulo op2 and put the result in rop. If the inverse exists, the return value is non-zero and rop will satisfy 0 ≤ rop < op2. If an inverse doesn’t exist the return value is zero and rop is undefined.

Function int mpz jacobi (mpz_t a, mpz_t b)

Calculate the Jacobi symbol (\frac{a}{b}). This is defined only for b odd.

Function int mpz legendre (mpz_t a, mpz_t p)

Calculate the Legendre symbol (\frac{a}{p}). This is defined only for p an odd positive prime, and for such p it’s identical to the Jacobi symbol.

Function int mpz kronecker (mpz_t a, mpz_t b) Function int mpz kronecker si (mpz_t a, long b)

Function int mpz kronecker ui (mpz_t a, unsigned long b)

Function int mpz si kronecker (long a, mpz_t b)

Function int mpz ui kronecker (unsigned long a, mpz_t b)

Calculate the Jacobi symbol (\frac{a}{b}) with the Kronecker extension (\frac{a}{2}) = (\frac{2}{a}) when a odd, or (\frac{a}{2}) = 0 when a even.

When b is odd the Jacobi symbol and Kronecker symbol are identical, so mpz_kronecker_ ui etc can be used for mixed precision Jacobi symbols too.

For more information see Henri Cohen section 1.4.2 (see Appendix B [References], page 109), or any number theory textbook. See also the example program ‘demos/qcn.c’

which uses mpz_kronecker_ui.
Function unsigned long int mpz remove (mpz_t rop, mpz_t op, mpz_t f )

Remove all occurrences of the factor f from op and store the result in rop. Return the multiplicity of f in op.

Function void mpz fac ui (mpz_t rop, unsigned long int op)

Set rop to op!, the factorial of op.

Function void mpz bin ui (mpz_t rop, mpz_t n, unsigned long int k)
Function void mpz bin uiui (mpz_t rop, unsigned long int n,
    unsigned long int k)

Compute the binomial coefficient \binom{n}{k} and store the result in rop. Negative values of n are supported by mpz_bin_ui, using the identity \binom{−n}{k} = (−1)^{k} \binom{n+k−1}{k}, see Knuth volume 1 section 1.2.6 part G.

Function void mpz fib ui (mpz_t fn, unsigned long int n)
Function void mpz fib2 ui (mpz_t fn, mpz_t fnsub1, unsigned long int n)

mpz_fib_ui sets fn to to F_{n}, the n’th Fibonacci number. mpz_fib2_ui sets fn to F_{n}, and fnsub1 to F_{n−1}.

These functions are designed for calculating isolated Fibonacci numbers. When a sequence of values is wanted it’s best to start with mpz_fib2_ui and iterate the defining F_{n+1} = F_{n} + F_{n−1} or similar.

Function void mpz lucnum ui (mpz_t ln, unsigned long int n)
Function void mpz lucnum2 ui (mpz_t ln, mpz_t lnsub1, unsigned long int n)

mpz_lucnum_ui sets ln to to L_{n}, the n’th Lucas number. mpz_lucnum2_ui sets ln to L_{n}, and lnsub1 to L_{n−1}.

These functions are designed for calculating isolated Lucas numbers. When a sequence of values is wanted it’s best to start with mpz_lucnum2_ui and iterate the defining L_{n+1} = L_{n} + L_{n−1} or similar.

The Fibonacci numbers and Lucas numbers are related sequences, so it’s never necessary to call both mpz_fib2_ui and mpz_lucnum2_ui. The formulas for going from Fibonacci to Lucas can be found in Section 16.7.4 [Lucas Numbers Algorithm], page 97, the reverse is straightforward too.

5.10 Comparison Functions

Function int mpz cmp (mpz_t op1, mpz_t op2) Function int mpz cmp d (mpz_t op1, double op2)

Macro int mpz cmp si (mpz_t op1, signed long int op2)
Macro int mpz cmp ui (mpz_t op1, unsigned long int op2)

Compare op1 and op2. Return a positive value if op1 > op2, zero if op1 = op2, or a negative value if op1 < op2.

Note that mpz_cmp_ui and mpz_cmp_si are macros and will evaluate their arguments more than once.

Function int mpz cmpabs (mpz_t op1, mpz_t op2) Function int mpz cmpabs d (mpz_t op1, double op2) Function int mpz cmpabs ui (mpz_t op1, unsigned long int op2)

Compare the absolute values of op1 and op2. Return a positive value if |op1| > |op2|, zero if |op1| = |op2|, or a negative value if |op1| < |op2|.

Note that mpz_cmpabs_si is a macro and will evaluate its arguments more than once.

Macro int mpz sgn (mpz_t op)

Return +1 if op > 0, 0 if op = 0, and −1 if op < 0.

This function is actually implemented as a macro. It evaluates its argument multiple times.

5.11 Logical and Bit Manipulation Functions

These functions behave as if twos complement arithmetic were used (although sign-magnitude is the actual implementation). The least significant bit is number 0.

Function void mpz and (mpz_t rop, mpz_t op1, mpz_t op2)

Set rop to op1 logical-and op2.

Function void mpz ior (mpz_t rop, mpz_t op1, mpz_t op2)

Set rop to op1 inclusive-or op2.

Function void mpz xor (mpz_t rop, mpz_t op1, mpz_t op2)

Set rop to op1 exclusive-or op2.

Function void mpz com (mpz_t rop, mpz_t op)

Set rop to the one’s complement of op.

Function unsigned long int mpz popcount (mpz_t op)

If op ≥ 0, return the population count of op, which is the number of 1 bits in the binary representation. If op < 0, the number of 1s is infinite, and the return value is MAX ULONG, the largest possible unsigned long.

Function unsigned long int mpz hamdist (mpz_t op1, mpz_t op2)

If op1 and op2 are both ≥ 0 or both < 0, return the hamming distance between the two operands, which is the number of bit positions where op1 and op2 have different bit values. If one operand is ≥ 0 and the other < 0 then the number of bits different is infinite, and the return value is MAX ULONG, the largest possible unsigned long.

Function unsigned long int mpz scan0 (mpz_t op, unsigned long int

starting bit)

Function unsigned long int mpz scan1 (mpz_t op, unsigned long int

starting bit) Scan op, starting from bit starting bit, towards more significant bits, until the first 0 or 1 bit (respectively) is found. Return the index of the found bit.

If the bit at starting bit is already what’s sought, then starting bit is returned.

If there’s no bit found, then MAX ULONG is returned. This will happen in mpz_scan0 past the end of a positive number, or mpz_scan1 past the end of a negative.

Function void mpz setbit (mpz_t rop, unsigned long int bit index)

Set bit bit index in rop.

Function void mpz clrbit (mpz_t rop, unsigned long int bit index)

Clear bit bit index in rop.

Function int mpz tstbit (mpz_t op, unsigned long int bit index)

Test bit bit index in op and return 0 or 1 accordingly.

5.12 Input and Output Functions

Functions that perform input from a stdio stream, and functions that output to a stdio stream. Passing a NULL pointer for a stream argument to any of these functions will make them read from stdin and write to stdout, respectively.

When using any of these functions, it is a good idea to include ‘stdio.h’ before ‘gmp.h’, since that will allow ‘gmp.h’ to define prototypes for these functions.

Function size_t mpz out str (FILE *stream, int base, mpz_t op)

Output op on stdio stream stream, as a string of digits in base base. The base may vary from 2 to 36.

Return the number of bytes written, or if an error occurred, return 0.

Function size_t mpz inp str (mpz_t rop, FILE *stream, int base)

Input a possibly white-space preceded string in base base from stdio stream stream, and put the read integer in rop. The base may vary from 2 to 36. If base is 0, the actual base is determined from the leading characters: if the first two characters are ‘0x’ or ‘0X’, hexadecimal is assumed, otherwise if the first character is ‘0’, octal is assumed, otherwise decimal is assumed.

Return the number of bytes read, or if an error occurred, return 0.

Function size_t mpz out raw (FILE *stream, mpz_t op)

Output op on stdio stream stream, in raw binary format. The integer is written in a portable format, with 4 bytes of size information, and that many bytes of limbs. Both the size and the limbs are written in decreasing significance order (i.e., in big-endian).

The output can be read with mpz_inp_raw.

Return the number of bytes written, or if an error occurred, return 0.

The output of this can not be read by mpz_inp_raw from GMP 1, because of changes necessary for compatibility between 32-bit and 64-bit machines.

Function size_t mpz inp raw (mpz_t rop, FILE *stream)

Input from stdio stream stream in the format written by mpz_out_raw, and put the result in rop. Return the number of bytes read, or if an error occurred, return 0.

This routine can read the output from mpz_out_raw also from GMP 1, in spite of changes necessary for compatibility between 32-bit and 64-bit machines.

5.13 Random Number Functions

The random number functions of GMP come in two groups; older function that rely on a global state, and newer functions that accept a state parameter that is read and modified. Please see the Chapter 9 [Random Number Functions], page 58 for more information on how to use and not to use random number functions.

Function void mpz urandomb (mpz_t rop, gmp_randstate_t state, unsigned
    long int n)

Generate a uniformly distributed random integer in the range 0 to 2^{n} − 1, inclusive.

The variable state must be initialized by calling one of the gmp_randinit functions (Section 9.1 [Random State Initialization], page 58) before invoking this function.

Function void mpz urandomm (mpz_t rop, gmp_randstate_t state, mpz_t n)

Generate a uniform random integer in the range 0 to n − 1, inclusive.

The variable state must be initialized by calling one of the gmp_randinit functions (Section 9.1 [Random State Initialization], page 58) before invoking this function.

Function void mpz rrandomb (mpz_t rop, gmp_randstate_t state, unsigned long
    int n)

Generate a random integer with long strings of zeros and ones in the binary representation. Useful for testing functions and algorithms, since this kind of random numbers have proven to be more likely to trigger corner-case bugs. The random number will be in the range 0 to 2^{n} − 1, inclusive.

The variable state must be initialized by calling one of the gmp_randinit functions (Section 9.1 [Random State Initialization], page 58) before invoking this function.

Function void mpz random (mpz_t rop, mp_size_t max size)

Generate a random integer of at most max size limbs. The generated random number doesn’t satisfy any particular requirements of randomness. Negative random numbers are generated when max size is negative.

This function is obsolete. Use mpz_urandomb or mpz_urandomm instead.

Function void mpz random2 (mpz_t rop, mp_size_t max size)

Generate a random integer of at most max size limbs, with long strings of zeros and ones in the binary representation. Useful for testing functions and algorithms, since this kind of random numbers have proven to be more likely to trigger corner-case bugs. Negative random numbers are generated when max size is negative.

This function is obsolete. Use mpz_rrandomb instead.

5.14 Integer Import and Export

mpz_t variables can be converted to and from arbitrary words of binary data with the following functions.

Function void mpz import (mpz_t rop, size_t count, int order, int size, int

    endian, size_t nails, const void *op)

Set rop from an array of word data at op.

The parameters specify the format of the data. count many words are read, each size bytes. order can be 1 for most significant word first or -1 for least significant first. Within each word endian can be 1 for most significant byte first, -1 for least significant first, or 0 for the native endianness of the host CPU. The most significant nails bits of each word are skipped, this can be 0 to use the full words.

There are no data alignment restrictions on op, any address is allowed.

Here’s an example converting an array of unsigned long data, most significant element first and host byte order within each value.

     unsigned long  a[20];
     mpz_t          z;
     mpz_import (z, 20, 1, sizeof(a[0]), 0, 0, a);

This example assumes the full sizeof bytes are used for data in the given type, which is usually true, and certainly true for unsigned long everywhere we know of. However on Cray vector systems it may be noted that short and int are always stored in 8 bytes (and with sizeof indicating that) but use only 32 or 46 bits. The nails feature can account for this, by passing for instance 8*sizeof(int)-INT_BIT.

Function void *mpz export (void *rop, size_t *count, int order, int size, int

endian, size_t nails, mpz_t op) Fill rop with word data from op.

The parameters specify the format of the data produced. Each word will be size bytes and order can be 1 for most significant word first or -1 for least significant first. Within each word endian can be 1 for most significant byte first, -1 for least significant first, or 0 for the native endianness of the host CPU. The most significant nails bits of each word are unused and set to zero, this can be 0 to produce full words.

The number of words produced is written to *count. rop must have enough space for the data, or if rop is NULL then a result array of the necessary size is allocated using the current GMP allocation function (see Chapter 14 [Custom Allocation], page 78). In either case the return value is the destination used, rop or the allocated block.

If op is non-zero then the most significant word produced will be non-zero. If op is zero then the count returned will be zero and nothing written to rop. If rop is NULL in this case, no block is allocated, just NULL is returned.

There are no data alignment restrictions on rop, any address is allowed. The sign of op is ignored, just the absolute value is used.

When an application is allocating space itself the required size can be determined with a calculation like the following. Since mpz_sizeinbase always returns at least 1, count here will be at least one, which avoids any portability problems with malloc(0), though if z is zero no space at all is actually needed.

     numb = 8*size - nail;
     count = (mpz_sizeinbase (z, 2) + numb-1) / numb;
     p = malloc (count * size);

5.15 Miscellaneous Functions

Function int mpz fits ulong p (mpz_t op) Function int mpz fits slong p (mpz_t op) Function int mpz fits uint p (mpz_t op) Function int mpz fits sint p (mpz_t op) Function int mpz fits ushort p (mpz_t op) Function int mpz fits sshort p (mpz_t op)

Return non-zero iff the value of op fits in an unsigned long int, signed long int,

unsigned int, signed int, unsigned short int, or signed short int, respectively.

Otherwise, return zero.

Macro int mpz odd p (mpz_t op) Macro int mpz even p (mpz_t op)

Determine whether op is odd or even, respectively. Return non-zero if yes, zero if no. These macros evaluate their argument more than once.

Function size_t mpz size (mpz_t op)

Return the size of op measured in number of limbs. If op is zero, the returned value will be zero.

Function size_t mpz sizeinbase (mpz_t op, int base)

Return the size of op measured in number of digits in base base. The base may vary from 2 to 36. The sign of op is ignored, just the absolute value is used. The result will be exact or 1 too big. If base is a power of 2, the result will always be exact. If op is zero the return value is always 1.

This function is useful in order to allocate the right amount of space before converting op to a string. The right amount of allocation is normally two more than the value returned by mpz_sizeinbase (one extra for a minus sign and one for the null-terminator).

6 Rational Number Functions

This chapter describes the GMP functions for performing arithmetic on rational numbers. These functions start with the prefix mpq_.

Rational numbers are stored in objects of type mpq_t.

All rational arithmetic functions assume operands have a canonical form, and canonicalize their result. The canonical from means that the denominator and the numerator have no common factors, and that the denominator is positive. Zero has the unique representation 0/1.

Pure assignment functions do not canonicalize the assigned variable. It is the responsibility of the user to canonicalize the assigned variable before any arithmetic operations are performed on that variable.

Function void mpq canonicalize (mpq_t op)

Remove any factors that are common to the numerator and denominator of op, and make the denominator positive.

6.1 Initialization and Assignment Functions

Function void mpq init (mpq_t dest rational)

Initialize dest rational and set it to 0/1. Each variable should normally only be initialized once, or at least cleared out (using the function mpq_clear) between each initialization.

Function void mpq clear (mpq_t rational number)

Free the space occupied by rational number. Make sure to call this function for all mpq_t variables when you are done with them.

Function void mpq set (mpq_t rop, mpq_t op) Function void mpq set z (mpq_t rop, mpz_t op)

Assign rop from op.

Function void mpq set ui (mpq_t rop, unsigned long int op1, unsigned long int

op2)

Function void mpq set si (mpq_t rop, signed long int op1, unsigned long int

op2) Set the value of rop to op1/op2. Note that if op1 and op2 have common factors, rop has to be passed to mpq_canonicalize before any operations are performed on rop.

Function int mpq set str (mpq_t rop, char *str, int base)

Set rop from a null-terminated string str in the given base.

The string can be an integer like “41” or a fraction like “41/152”. The fraction must be in canonical form (see Chapter 6 [Rational Number Functions], page 39), or if not then

mpq_canonicalize must be called.

The numerator and optional denominator are parsed the same as in mpz_set_str (see Section 5.2 [Assigning Integers], page 26). White space is allowed in the string, and is simply ignored. The base can vary from 2 to 36, or if base is 0 then the leading characters are used: 0x for hex, 0 for octal, or decimal otherwise. Note that this is done separately for the numerator and denominator, so for instance 0xEF/100 is 239/100, whereas 0xEF/0x100 is 239/256.

The return value is 0 if the entire string is a valid number, or −1 if not.

Function void mpq swap (mpq_t rop1, mpq_t rop2)

Swap the values rop1 and rop2 efficiently.

6.2 Conversion Functions

Function double mpq get d (mpq_t op)

Convert op to a double.

Function void mpq set d (mpq_t rop, double op) Function void mpq set f (mpq_t rop, mpf_t op)

Set rop to the value of op, without rounding.

Function char * mpq get str (char *str, int base, mpq_t op)

Convert op to a string of digits in base base. The base may vary from 2 to 36. The string will be of the form ‘num/den’, or if the denominator is 1 then just ‘num’.

If str is NULL, the result string is allocated using the current allocation function (see Chapter 14 [Custom Allocation], page 78). The block will be strlen(str)+1 bytes, that being exactly enough for the string and null-terminator.

If str is not NULL, it should point to a block of storage large enough for the result, that being

     mpz_sizeinbase (mpq_numref(op), base)
     + mpz_sizeinbase (mpq_denref(op), base) + 3

The three extra bytes are for a possible minus sign, possible slash, and the null-terminator.

A pointer to the result string is returned, being either the allocated block, or the given str.

6.3 Arithmetic Functions

Function void mpq add (mpq_t sum, mpq_t addend1, mpq_t addend2)

Set sum to addend1 + addend2.

Function void mpq sub (mpq_t difference, mpq_t minuend, mpq_t subtrahend)

Set difference to minuend − subtrahend.

Function void mpq mul (mpq_t product, mpq_t multiplier, mpq_t multiplicand)

Set product to multiplier × multiplicand.

Function void mpq mul 2exp (mpq_t rop, mpq_t op1, unsigned long int op2)

Set rop to op1 × 2^{op2}.

Function void mpq div (mpq_t quotient, mpq_t dividend, mpq_t divisor)

Set quotient to dividend/divisor.

Function void mpq div 2exp (mpq_t rop, mpq_t op1, unsigned long int op2)

Set rop to op1/2^{op2}.

Function void mpq neg (mpq_t negated operand, mpq_t operand)

Set negated operand to −operand.

Function void mpq abs (mpq_t rop, mpq_t op)

Set rop to the absolute value of op.

Function void mpq inv (mpq_t inverted number, mpq_t number)

Set inverted number to 1/number. If the new denominator is zero, this routine will divide by zero.

6.4 Comparison Functions

Function int mpq cmp (mpq_t op1, mpq_t op2)

Compare op1 and op2. Return a positive value if op1 > op2, zero if op1 = op2, and a negative value if op1 < op2.

To determine if two rationals are equal, mpq_equal is faster than mpq_cmp.

Macro int mpq cmp ui (mpq_t op1, unsigned long int num2, unsigned long int

den2)

Macro int mpq cmp si (mpq_t op1, long int num2, unsigned long int den2)

Compare op1 and num2/den2. Return a positive value if op1 > num2/den2, zero if op1 = num2/den2, and a negative value if op1 < num2/den2.

num2 and den2 are allowed to have common factors.

These functions are implemented as a macros and evaluate their arguments multiple times.

Macro int mpq sgn (mpq_t op)

Return +1 if op > 0, 0 if op = 0, and −1 if op < 0.

This function is actually implemented as a macro. It evaluates its arguments multiple times.

Function int mpq equal (mpq_t op1, mpq_t op2)

Return non-zero if op1 and op2 are equal, zero if they are non-equal. Although mpq_cmp can be used for the same purpose, this function is much faster.

6.5 Applying Integer Functions to Rationals

The set of mpq functions is quite small. In particular, there are few functions for either input or output. The following functions give direct access to the numerator and denominator of an

mpq_t.

Note that if an assignment to the numerator and/or denominator could take an mpq_t out of the canonical form described at the start of this chapter (see Chapter 6 [Rational Number Functions], page 39) then mpq_canonicalize must be called before any other mpq functions are applied to that mpq_t.

Macro mpz_t mpq numref (mpq_t op) Macro mpz_t mpq denref (mpq_t op)

Return a reference to the numerator and denominator of op, respectively. The mpz functions can be used on the result of these macros.

Function void mpq get num (mpz_t numerator, mpq_t rational) Function void mpq get den (mpz_t denominator, mpq_t rational) Function void mpq set num (mpq_t rational, mpz_t numerator) Function void mpq set den (mpq_t rational, mpz_t denominator)

Get or set the numerator or denominator of a rational. These functions are equivalent to calling mpz_set with an appropriate mpq_numref or mpq_denref. Direct use of mpq_ numref or mpq_denref is recommended instead of these functions.

6.6 Input and Output Functions

When using any of these functions, it’s a good idea to include ‘stdio.h’ before ‘gmp.h’, since that will allow ‘gmp.h’ to define prototypes for these functions.

Passing a NULL pointer for a stream argument to any of these functions will make them read from stdin and write to stdout, respectively.

Function size_t mpq out str (FILE *stream, int base, mpq_t op)

Output op on stdio stream stream, as a string of digits in base base. The base may vary from 2 to 36. Output is in the form ‘num/den’ or if the denominator is 1 then just ‘num’.

Return the number of bytes written, or if an error occurred, return 0.

Function size_t mpq inp str (mpq_t rop, FILE *stream, int base)

Read a string of digits from stream and convert them to a rational in rop. Any initial white-space characters are read and discarded. Return the number of characters read (including white space), or 0 if a rational could not be read.

The input can be a fraction like ‘17/63’ or just an integer like ‘123’. Reading stops at the first character not in this form, and white space is not permitted within the string. If the input might not be in canonical form, then mpq_canonicalize must be called (see Chapter 6 [Rational Number Functions], page 39).

The base can be between 2 and 36, or can be 0 in which case the leading characters of the string determine the base, ‘0x’ or ‘0X’ for hexadecimal, ‘0’ for octal, or decimal otherwise. The leading characters are examined separately for the numerator and denominator of a fraction, so for instance ‘0x10/11’ is 16/11, whereas ‘0x10/0x11’ is 16/17.

7 Floating-point Functions

GMP floating point numbers are stored in objects of type mpf_t and functions operating on them have an mpf_prefix.

The mantissa of each float has a user-selectable precision, limited only by available memory. Each variable has its own precision, and that can be increased or decreased at any time.

The exponent of each float is a fixed precision, one machine word on most systems. In the current implementation the exponent is a count of limbs, so for example on a 32-bit system this means a range of roughly 2^{−68719476768} to 2^{68719476736}, or on a 64-bit system this will be greater. Note however mpf_get_str can only return an exponent which fits an mp_exp_t and currently mpf_set_str doesn’t accept exponents bigger than a long.

Each variable keeps a size for the mantissa data actually in use. This means that if a float is exactly represented in only a few bits then only those bits will be used in a calculation, even if the selected precision is high.

All calculations are performed to the precision of the destination variable. Each function is defined to calculate with “infinite precision” followed by a truncation to the destination precision, but of course the work done is only what’s needed to determine a result under that definition.

The precision selected for a variable is a minimum value, GMP may increase it a little to facilitate efficient calculation. Currently this means rounding up to a whole limb, and then sometimes having a further partial limb, depending on the high limb of the mantissa. But applications shouldn’t be concerned by such details.

mpf functions and variables have no special notion of infinity or not-a-number, and applications must take care not to overflow the exponent or results will be unpredictable. This might change in a future release.

Note that the mpf functions are not intended as a smooth extension to IEEE P754 arithmetic. In particular results obtained on one computer often differ from the results on a computer with a different word size.

7.1 Initialization Functions

Function void mpf set default prec (unsigned long int prec)

Set the default precision to be at least prec bits. All subsequent calls to mpf_init will use this precision, but previously initialized variables are unaffected.

Function unsigned long int mpf get default prec (void)

Return the default default precision actually used.

An mpf_t object must be initialized before storing the first value in it. The functions mpf_init and mpf_init2 are used for that purpose.

Function void mpf init (mpf_t x)

Initialize x to 0. Normally, a variable should be initialized once only or at least be cleared, using mpf_clear, between initializations. The precision of x is undefined unless a default precision has already been established by a call to mpf_set_default_prec.

Function void mpf init2 (mpf_t x, unsigned long int prec)

Initialize x to 0 and set its precision to be at least prec bits. Normally, a variable should be initialized once only or at least be cleared, using mpf_clear, between initializations.

Function void mpf clear (mpf_t x)

Free the space occupied by x. Make sure to call this function for all mpf_t variables when you are done with them.

Here is an example on how to initialize floating-point variables:

{
  mpf_t x, y;
  mpf_init (x);           /* use default precision */
  mpf_init2 (y, 256);     /* precision at least 256 bits */
  ...
  /* Unless the program is about to exit, do ... */
  mpf_clear (x);
  mpf_clear (y);
}

The following three functions are useful for changing the precision during a calculation. A typical use would be for adjusting the precision gradually in iterative algorithms like Newton-Raphson, making the computation precision closely match the actual accurate part of the numbers.

Function unsigned long int mpf get prec (mpf_t op)

Return the current precision of op, in bits.

Function void mpf set prec (mpf_t rop, unsigned long int prec)

Set the precision of rop to be at least prec bits. The value in rop will be truncated to the new precision.

This function requires a call to realloc, and so should not be used in a tight loop.

Function void mpf set prec raw (mpf_t rop, unsigned long int prec)

Set the precision of rop to be at least prec bits, without changing the memory allocated.

prec must be no more than the allocated precision for rop, that being the precision when rop was initialized, or in the most recent mpf_set_prec.

The value in rop is unchanged, and in particular if it had a higher precision than prec it will retain that higher precision. New values written to rop will use the new prec.

Before calling mpf_clear or the full mpf_set_prec, another mpf_set_prec_raw call must be made to restore rop to its original allocated precision. Failing to do so will have unpredictable results.

mpf_get_prec can be used before mpf_set_prec_raw to get the original allocated precision. After mpf_set_prec_raw it reflects the prec value set.

mpf_set_prec_raw is an efficient way to use an mpf_t variable at different precisions during a calculation, perhaps to gradually increase precision in an iteration, or just to use various different precisions for different purposes during a calculation.

7.2 Assignment Functions

These functions assign new values to already initialized floats (see Section 7.1 [Initializing Floats], page 43).

Function void mpf set (mpf_t rop, mpf_t op)

Function void mpf set ui (mpf_t rop, unsigned long int op)
Function void mpf set si (mpf_t rop, signed long int op)

Function void mpf set d (mpf_t rop, double op) Function void mpf set z (mpf_t rop, mpz_t op) Function void mpf set q (mpf_t rop, mpq_t op)

Set the value of rop from op.

Function int mpf set str (mpf_t rop, char *str, int base)

Set the value of rop from the string in str. The string is of the form ‘M@N’ or, if the base is 10 or less, alternatively ‘MeN’. ‘M’ is the mantissa and ‘N’ is the exponent. The mantissa is always in the specified base. The exponent is either in the specified base or, if base is negative, in decimal. The decimal point expected is taken from the current locale, on systems providing localeconv.

The argument base may be in the ranges 2 to 36, or −36 to −2. Negative values are used to specify that the exponent is in decimal.

Unlike the corresponding mpz function, the base will not be determined from the leading characters of the string if base is 0. This is so that numbers like ‘0.23’ are not interpreted as octal.

White space is allowed in the string, and is simply ignored. [This is not really true; whitespace is ignored in the beginning of the string and within the mantissa, but not in other places, such as after a minus sign or in the exponent. We are considering changing the definition of this function, making it fail when there is any white-space in the input, since that makes a lot of sense. Please tell us your opinion about this change. Do you really want it to accept "3 14" as meaning 314 as it does now?]

This function returns 0 if the entire string is a valid number in base base. Otherwise it returns −1.

Function void mpf swap (mpf_t rop1, mpf_t rop2)

Swap rop1 and rop2 efficiently. Both the values and the precisions of the two variables are swapped.

7.3 Combined Initialization and Assignment Functions

For convenience, GMP provides a parallel series of initialize-and-set functions which initialize the output and then store the value there. These functions’ names have the form mpf_init_set...

Once the float has been initialized by any of the mpf_init_set... functions, it can be used as the source or destination operand for the ordinary float functions. Don’t use an initialize-and-set function on a variable already initialized!

Function void mpf init set (mpf_t rop, mpf_t op) Function void mpf init set ui (mpf_t rop, unsigned long int op) Function void mpf init set si (mpf_t rop, signed long int op) Function void mpf init set d (mpf_t rop, double op)

Initialize rop and set its value from op.

The precision of rop will be taken from the active default precision, as set by mpf_set_

default_prec.

Function int mpf init set str (mpf_t rop, char *str, int base)

Initialize rop and set its value from the string in str. See mpf_set_str above for details on the assignment operation.

Note that rop is initialized even if an error occurs. (I.e., you have to call mpf_clear for it.)

The precision of rop will be taken from the active default precision, as set by mpf_set_

default_prec.

7.4 Conversion Functions

Function double mpf get d (mpf_t op)

Convert op to a double.

Function double mpf get d 2exp (signed long int exp, mpf_t op)

Find d and exp such that d × 2^{exp}, with 0.5 ≤ |d| < 1, is a good approximation to op. This is similar to the standard C function frexp.

Function long mpf get si (mpf_t op) Function unsigned long mpf get ui (mpf_t op)

Convert op to a long or unsigned long, truncating any fraction part. If op is too big for the return type, the result is undefined.

See also mpf_fits_slong_p and mpf_fits_ulong_p (see Section 7.8 [Miscellaneous Float Functions], page 49).

Function char * mpf get str (char *str, mp_exp_t *expptr, int base, size_t

n digits, mpf_t op) Convert op to a string of digits in base base. base can be 2 to 36. Up to n digits digits will be generated. Trailing zeros are not returned. No more digits than can be accurately represented by op are ever generated. If n digits is 0 then that accurate maximum number of digits are generated.

If str is NULL, the result string is allocated using the current allocation function (see Chapter 14 [Custom Allocation], page 78). The block will be strlen(str)+1 bytes, that being exactly enough for the string and null-terminator.

If str is not NULL, it should point to a block of n digits + 2 bytes, that being enough for the mantissa, a possible minus sign, and a null-terminator. When n digits is 0 to get all significant digits, an application won’t be able to know the space required, and str should be NULL in that case.

The generated string is a fraction, with an implicit radix point immediately to the left of the first digit. The applicable exponent is written through the expptr pointer. For example, the number 3.1416 would be returned as string "31416" and exponent 1.

When op is zero, an empty string is produced and the exponent returned is 0.

A pointer to the result string is returned, being either the allocated block or the given str.

7.5 Arithmetic Functions

Function void mpf add (mpf_t rop, mpf_t op1, mpf_t op2)

Function void mpf add ui (mpf_t rop, mpf_t op1, unsigned long int op2)

Set rop to op1 + op2.

Function void mpf sub (mpf_t rop, mpf_t op1, mpf_t op2)

Function void mpf ui sub (mpf_t rop, unsigned long int op1, mpf_t op2)
Function void mpf sub ui (mpf_t rop, mpf_t op1, unsigned long int op2)

Set rop to op1 − op2.

Function void mpf mul (mpf_t rop, mpf_t op1, mpf_t op2)

Function void mpf mul ui (mpf_t rop, mpf_t op1, unsigned long int op2)

Set rop to op1 × op2.

Division is undefined if the divisor is zero, and passing a zero divisor to the divide functions will make these functions intentionally divide by zero. This lets the user handle arithmetic exceptions in these functions in the same manner as other arithmetic exceptions.

Function void mpf div (mpf_t rop, mpf_t op1, mpf_t op2)

Function void mpf ui div (mpf_t rop, unsigned long int op1, mpf_t op2)
Function void mpf div ui (mpf_t rop, mpf_t op1, unsigned long int op2)

Set rop to op1/op2.

Function void mpf sqrt (mpf_t rop, mpf_t op)

Function void mpf sqrt ui (mpf_t rop, unsigned long int op)

Set rop to \sqrt{op}.

Function void mpf pow ui (mpf_t rop, mpf_t op1, unsigned long int op2)

Set rop to op1^{op2}.

Function void mpf neg (mpf_t rop, mpf_t op)

Set rop to −op.

Function void mpf abs (mpf_t rop, mpf_t op)

Set rop to the absolute value of op.

Function void mpf mul 2exp (mpf_t rop, mpf_t op1, unsigned long int op2)

Set rop to op1 × 2^{op2}.

Function void mpf div 2exp (mpf_t rop, mpf_t op1, unsigned long int op2)

Set rop to op1/2^{op2}.

7.6 Comparison Functions

Function int mpf cmp (mpf_t op1, mpf_t op2) Function int mpf cmp d (mpf_t op1, double op2)

Function int mpf cmp ui (mpf_t op1, unsigned long int op2)

Function int mpf cmp si (mpf_t op1, signed long int op2)

Compare op1 and op2. Return a positive value if op1 > op2, zero if op1 = op2, and a negative value if op1 < op2.

Function int mpf eq (mpf_t op1, mpf_t op2, unsigned long int op3)

Return non-zero if the first op3 bits of op1 and op2 are equal, zero otherwise. I.e., test of op1 and op2 are approximately equal.

Caution: Currently only whole limbs are compared, and only in an exact fashion. In the future values like 1000 and 0111 may be considered the same to 3 bits (on the basis that their difference is that small).

Function void mpf reldiff (mpf_t rop, mpf_t op1, mpf_t op2)

Compute the relative difference between op1 and op2 and store the result in rop. This is |op1 − op2|/op1.

Macro int mpf sgn (mpf_t op)

Return +1 if op > 0, 0 if op = 0, and −1 if op < 0.

This function is actually implemented as a macro. It evaluates its arguments multiple times.

7.7 Input and Output Functions

Functions that perform input from a stdio stream, and functions that output to a stdio stream. Passing a NULL pointer for a stream argument to any of these functions will make them read from stdin and write to stdout, respectively.

When using any of these functions, it is a good idea to include ‘stdio.h’ before ‘gmp.h’, since that will allow ‘gmp.h’ to define prototypes for these functions.

Function size_t mpf out str (FILE *stream, int base, size_t n digits, mpf_t op)

Print op to stream, as a string of digits. Return the number of bytes written, or if an error occurred, return 0.

The mantissa is prefixed with an ‘0.’ and is in the given base, which may vary from 2 to 36. An exponent then printed, separated by an ‘e’, or if base is greater than 10 then by an ‘@’. The exponent is always in decimal. The decimal point follows the current locale, on systems providing localeconv.

Up to n digits will be printed from the mantissa, except that no more digits than are accurately representable by op will be printed. n digits can be 0 to select that accurate maximum.

Function size_t mpf inp str (mpf_t rop, FILE *stream, int base)

Read a string in base base from stream, and put the read float in rop. The string is of the form ‘M@N’ or, if the base is 10 or less, alternatively ‘MeN’. ‘M’ is the mantissa and ‘N’ is the exponent. The mantissa is always in the specified base. The exponent is either in the specified base or, if base is negative, in decimal. The decimal point expected is taken from the current locale, on systems providing localeconv.

The argument base may be in the ranges 2 to 36, or −36 to −2. Negative values are used to specify that the exponent is in decimal.

Unlike the corresponding mpz function, the base will not be determined from the leading characters of the string if base is 0. This is so that numbers like ‘0.23’ are not interpreted as octal.

Return the number of bytes read, or if an error occurred, return 0.

7.8 Miscellaneous Functions

Function void mpf ceil (mpf_t rop, mpf_t op) Function void mpf floor (mpf_t rop, mpf_t op) Function void mpf trunc (mpf_t rop, mpf_t op)

Set rop to op rounded to an integer. mpf_ceil rounds to the next higher integer, mpf_ floor to the next lower, and mpf_trunc to the integer towards zero.

Function int mpf integer p (mpf_t op)

Return non-zero if op is an integer.

Function int mpf fits ulong p (mpf_t op) Function int mpf fits slong p (mpf_t op) Function int mpf fits uint p (mpf_t op) Function int mpf fits sint p (mpf_t op) Function int mpf fits ushort p (mpf_t op) Function int mpf fits sshort p (mpf_t op)

Return non-zero if op would fit in the respective C data type, when truncated to an integer.

Function void mpf urandomb (mpf_t rop, gmp_randstate_t state, unsigned long

int nbits) Generate a uniformly distributed random float in rop, such that 0 ≤ rop < 1, with nbits significant bits in the mantissa.

The variable state must be initialized by calling one of the gmp_randinit functions (Section 9.1 [Random State Initialization], page 58) before invoking this function.

Function void mpf random2 (mpf_t rop, mp_size_t max size, mp_exp_t exp)

Generate a random float of at most max size limbs, with long strings of zeros and ones in the binary representation. The exponent of the number is in the interval −exp to exp. This function is useful for testing functions and algorithms, since this kind of random numbers have proven to be more likely to trigger corner-case bugs. Negative random numbers are generated when max size is negative.

8 Low-level Functions

This chapter describes low-level GMP functions, used to implement the high-level GMP functions, but also intended for time-critical user code.

These functions start with the prefix mpn_.

The mpn functions are designed to be as fast as possible, not to provide a coherent calling interface. The different functions have somewhat similar interfaces, but there are variations that make them hard to use. These functions do as little as possible apart from the real multiple precision computation, so that no time is spent on things that not all callers need.

A source operand is specified by a pointer to the least significant limb and a limb count. A destination operand is specified by just a pointer. It is the responsibility of the caller to ensure that the destination has enough space for storing the result.

With this way of specifying operands, it is possible to perform computations on subranges of an argument, and store the result into a subrange of a destination.

A common requirement for all functions is that each source area needs at least one limb. No size argument may be zero. Unless otherwise stated, in-place operations are allowed where source and destination are the same, but not where they only partly overlap.

The mpn functions are the base for the implementation of the mpz_, mpf_, and mpq_functions.

This example adds the number beginning at s1p and the number beginning at s2p and writes the sum at destp. All areas have n limbs.

     cy = mpn_add_n (destp, s1p, s2p, n)

In the notation used here, a source operand is identified by the pointer to the least significant limb, and the limb count in braces. For example, {s1p, s1n}.

Function mp_limb_t mpn add n (mp_limb_t *rp, const mp_limb_t *s1p, const
         mp_limb_t *s2p, mp_size_t n)

Add {s1p, n} and {s2p, n}, and write the n least significant limbs of the result to rp. Return carry, either 0 or 1.

This is the lowest-level function for addition. It is the preferred function for addition, since it is written in assembly for most CPUs. For addition of a variable to itself (i.e., s1p equals s2p, use mpn_lshift with a count of 1 for optimal speed.

Function mp_limb_t mpn add 1 (mp_limb_t *rp, const mp_limb_t *s1p,
         mp_size_t n, mp_limb_t s2limb)

Add {s1p, n} and s2limb, and write the n least significant limbs of the result to rp. Return carry, either 0 or 1.

Function mp_limb_t mpn add (mp_limb_t *rp, const mp_limb_t *s1p, mp_size_t
         s1n, const mp_limb_t *s2p, mp_size_t s2n)

Add {s1p, s1n} and {s2p, s2n}, and write the s1n least significant limbs of the result to rp. Return carry, either 0 or 1.

This function requires that s1n is greater than or equal to s2n.

Function mp_limb_t mpn sub n (mp_limb_t *rp, const mp_limb_t *s1p, const
    mp_limb_t *s2p, mp_size_t n)

Subtract {s2p, n} from {s1p, n}, and write the n least significant limbs of the result to rp. Return borrow, either 0 or 1.

This is the lowest-level function for subtraction. It is the preferred function for subtraction, since it is written in assembly for most CPUs.

Function mp_limb_t mpn sub 1 (mp_limb_t *rp, const mp_limb_t *s1p,
    mp_size_t n, mp_limb_t s2limb)

Subtract s2limb from {s1p, n}, and write the n least significant limbs of the result to rp. Return borrow, either 0 or 1.

Function mp_limb_t mpn sub (mp_limb_t *rp, const mp_limb_t *s1p, mp_size_t
    s1n, const mp_limb_t *s2p, mp_size_t s2n)

Subtract {s2p, s2n} from {s1p, s1n}, and write the s1n least significant limbs of the result to rp. Return borrow, either 0 or 1.

This function requires that s1n is greater than or equal to s2n.

Function void mpn mul n (mp_limb_t *rp, const mp_limb_t *s1p, const
    mp_limb_t *s2p, mp_size_t n)

Multiply {s1p, n} and {s2p, n}, and write the 2*n-limb result to rp.

The destination has to have space for 2*n limbs, even if the product’s most significant limb is zero.

Function mp_limb_t mpn mul 1 (mp_limb_t *rp, const mp_limb_t *s1p,
    mp_size_t n, mp_limb_t s2limb)

Multiply {s1p, n} by s2limb, and write the n least significant limbs of the product to rp. Return the most significant limb of the product. {s1p, n} and {rp, n} are allowed to overlap provided rp ≤ s1p.

This is a low-level function that is a building block for general multiplication as well as other operations in GMP. It is written in assembly for most CPUs.

Don’t call this function if s2limb is a power of 2; use mpn_lshift with a count equal to the logarithm of s2limb instead, for optimal speed.

Function mp_limb_t mpn addmul 1 (mp_limb_t *rp, const mp_limb_t *s1p,
    mp_size_t n, mp_limb_t s2limb)

Multiply {s1p, n} and s2limb, and add the n least significant limbs of the product to {rp, n} and write the result to rp. Return the most significant limb of the product, plus carry-out from the addition.

This is a low-level function that is a building block for general multiplication as well as other operations in GMP. It is written in assembly for most CPUs.

Function mp_limb_t mpn submul 1 (mp_limb_t *rp, const mp_limb_t *s1p,
    mp_size_t n, mp_limb_t s2limb)

Multiply {s1p, n} and s2limb, and subtract the n least significant limbs of the product from {rp, n} and write the result to rp. Return the most significant limb of the product, minus borrow-out from the subtraction.

This is a low-level function that is a building block for general multiplication and division as well as other operations in GMP. It is written in assembly for most CPUs.

Function mp_limb_t mpn mul (mp_limb_t *rp, const mp_limb_t *s1p, mp_size_t
    s1n, const mp_limb_t *s2p, mp_size_t s2n)

Multiply {s1p, s1n} and {s2p, s2n}, and write the result to rp. Return the most significant limb of the result.

The destination has to have space for s1n + s2n limbs, even if the result might be one limb smaller.

This function requires that s1n is greater than or equal to s2n. The destination must be distinct from both input operands.

Function void mpn tdiv qr (mp_limb_t *qp, mp_limb_t *rp, mp_size_t qxn,
    const mp_limb_t *np, mp_size_t nn, const mp_limb_t *dp, mp_size_t dn)

Divide {np, nn} by {dp, dn} and put the quotient at {qp, nn−dn+1} and the remainder at {rp, dn}. The quotient is rounded towards 0.

No overlap is permitted between arguments. nn must be greater than or equal to dn. The most significant limb of dp must be non-zero. The qxn operand must be zero.

Function mp_limb_t mpn divrem (mp_limb_t *r1p, mp_size_t qxn, mp_limb_t
    *rs2p, mp_size_t rs2n, const mp_limb_t *s3p, mp_size_t s3n)

[This function is obsolete. Please call mpn_tdiv_qr instead for best performance.]

Divide {rs2p, rs2n} by {s3p, s3n}, and write the quotient at r1p, with the exception of the most significant limb, which is returned. The remainder replaces the dividend at rs2p; it will be s3n limbs long (i.e., as many limbs as the divisor).

In addition to an integer quotient, qxn fraction limbs are developed, and stored after the integral limbs. For most usages, qxn will be zero.

It is required that rs2n is greater than or equal to s3n. It is required that the most significant bit of the divisor is set.

If the quotient is not needed, pass rs2p + s3n as r1p. Aside from that special case, no overlap between arguments is permitted.

Return the most significant limb of the quotient, either 0 or 1.

The area at r1p needs to be rs2n − s3n + qxn limbs large.

Function mp_limb_t mpn divrem 1 (mp_limb_t *r1p, mp_size_t qxn,
    mp_limb_t *s2p, mp_size_t s2n, mp_limb_t s3limb)
Macro mp_limb_t mpn divmod 1 (mp_limb_t *r1p, mp_limb_t *s2p,
    mp_size_t s2n, mp_limb_t s3limb)

Divide {s2p, s2n} by s3limb, and write the quotient at r1p. Return the remainder.

The integer quotient is written to {r1p+qxn, s2n} and in addition qxn fraction limbs are developed and written to {r1p, qxn}. Either or both s2n and qxn can be zero. For most usages, qxn will be zero.

mpn_divmod_1 exists for upward source compatibility and is simply a macro calling mpn_ divrem_1 with a qxn of 0.

The areas at r1p and s2p have to be identical or completely separate, not partially overlapping.

Function mp_limb_t mpn divmod (mp_limb_t *r1p, mp_limb_t *rs2p, mp_size_t
    rs2n, const mp_limb_t *s3p, mp_size_t s3n)

[This function is obsolete. Please call mpn_tdiv_qr instead for best performance.]

Macro mp_limb_t mpn divexact by3 (mp_limb_t *rp, mp_limb_t *sp,
    mp_size_t n)
Function mp_limb_t mpn divexact by3c (mp_limb_t *rp, mp_limb_t *sp,
    mp_size_t n, mp_limb_t carry)

Divide {sp, n} by 3, expecting it to divide exactly, and writing the result to {rp, n}. If 3 divides exactly, the return value is zero and the result is the quotient. If not, the return value is non-zero and the result won’t be anything useful.

mpn_divexact_by3c takes an initial carry parameter, which can be the return value from a previous call, so a large calculation can be done piece by piece from low to high. mpn_ divexact_by3 is simply a macro calling mpn_divexact_by3c with a 0 carry parameter.

These routines use a multiply-by-inverse and will be faster than mpn_divrem_1 on CPUs with fast multiplication but slow division.

The source a, result q, size n, initial carry i, and return value c satisfy cb^{n} + a − i = 3q, where b = 2mp bits per limb. The return c is always 0, 1 or 2, and the initial carry i must also be 0, 1 or 2 (these are both borrows really). When c = 0 clearly q = (a − i)/3. When c ≠ 0, the remainder (a − i) mod 3 is given by 3 − c, because b ≡ 1 mod 3 (when mp_bits_per_limb is even, which is always so currently).

Function mp_limb_t mpn mod 1 (mp_limb_t *s1p, mp_size_t s1n, mp_limb_t

s2limb) Divide {s1p, s1n} by s2limb, and return the remainder. s1n can be zero.

Function mp_limb_t mpn bdivmod (mp_limb_t *rp, mp_limb_t *s1p, mp_size_t
    s1n, const mp_limb_t *s2p, mp_size_t s2n, unsigned long int d)

This function puts the low ⌊d/mp bits per limb⌋ limbs of q = {s1p, s1n}/{s2p, s2n} mod 2^{d} at rp, and returns the high d mod mp_bits_per_limb bits of q.

{s1p, s1n} - q * {s2p, s2n} mod 2s1n*mp bits per limb is placed at s1p. Since the low ⌊d/mp bits per limb⌋ limbs of this difference are zero, it is possible to overwrite the low limbs at s1p with this difference, provided rp ≤ s1p.

This function requires that s1n ∗ mp bits per limb ≥ D, and that {s2p, s2n} is odd.

This interface is preliminary. It might change incompatibly in future revisions.

Function mp_limb_t mpn lshift (mp_limb_t *rp, const mp_limb_t *sp, mp_size_t
    n, unsigned int count)

Shift {sp, n} left by count bits, and write the result to {rp, n}. The bits shifted out at the left are returned in the least significant count bits of the return value (the rest of the return value is zero).

count must be in the range 1 to mp_bits_per_limb−1. The regions {sp, n} and {rp, n} may overlap, provided rp ≥ sp.

This function is written in assembly for most CPUs.

Function mp_limb_t mpn rshift (mp_limb_t *rp, const mp_limb_t *sp, mp_size_t
    n, unsigned int count)

Shift {sp, n} right by count bits, and write the result to {rp, n}. The bits shifted out at the right are returned in the most significant count bits of the return value (the rest of the return value is zero).

count must be in the range 1 to mp_bits_per_limb−1. The regions {sp, n} and {rp, n} may overlap, provided rp ≤ sp.

This function is written in assembly for most CPUs.

Function int mpn cmp (const mp_limb_t *s1p, const mp_limb_t *s2p, mp_size_t

n) Compare {s1p, n} and {s2p, n} and return a positive value if s1 > s2, 0 if they are equal, or a negative value if s1 < s2.

Function mp_size_t mpn gcd (mp_limb_t *rp, mp_limb_t *s1p, mp_size_t s1n,
    mp_limb_t *s2p, mp_size_t s2n)

Set {rp, retval} to the greatest common divisor of {s1p, s1n} and {s2p, s2n}. The result can be up to s2n limbs, the return value is the actual number produced. Both source operands are destroyed.

{s1p, s1n} must have at least as many bits as {s2p, s2n}. {s2p, s2n} must be odd. Both operands must have non-zero most significant limbs. No overlap is permitted between {s1p, s1n} and {s2p, s2n}.

Function mp_limb_t mpn gcd 1 (const mp_limb_t *s1p, mp_size_t s1n,
    mp_limb_t s2limb)

Return the greatest common divisor of {s1p, s1n} and s2limb. Both operands must be non-zero.

Function mp_size_t mpn gcdext (mp_limb_t *r1p, mp_limb_t *r2p, mp_size_t
    *r2n, mp_limb_t *s1p, mp_size_t s1n, mp_limb_t *s2p, mp_size_t s2n)

Calculate the greatest common divisor of {s1p, s1n} and {s2p, s2n}. Store the gcd at {r1p, retval} and the first cofactor at {r2p, *r2n}, with *r2n negative if the cofactor is negative. r1p and r2p should each have room for s1n + 1 limbs, but the return value and value stored through r2n indicate the actual number produced.

{s1p, s1n} ≥ {s2p, s2n} is required, and both must be non-zero. The regions {s1p, s1n+1} and {s2p, s2n + 1} are destroyed (i.e. the operands plus an extra limb past the end of each).

The cofactor r1 will satisfy r_{2}s_{1} + ks_{2} = r_{1}. The second cofactor k is not calculated but can easily be obtained from (r_{1} − r_{2}s_{1})/s_{2}.

Function mp_size_t mpn sqrtrem (mp_limb_t *r1p, mp_limb_t *r2p, const
    mp_limb_t *sp, mp_size_t n)

Compute the square root of {sp, n} and put the result at {r1p, ⌈n/2⌉} and the remainder at {r2p, retval}. r2p needs space for n limbs, but the return value indicates how many are produced.

The most significant limb of {sp, n} must be non-zero. The areas {r1p, ⌈n/2⌉} and {sp, n} must be completely separate. The areas {r2p, n} and {sp, n} must be either identical or completely separate.

If the remainder is not wanted then r2p can be NULL, and in this case the return value is zero or non-zero according to whether the remainder would have been zero or non-zero.

A return value of zero indicates a perfect square. See also mpz_perfect_square_p.

Function mp_size_t mpn get str (unsigned char *str, int base, mp_limb_t *s1p,
    mp_size_t s1n)

Convert {s1p, s1n} to a raw unsigned char array at str in base base, and return the number of characters produced. There may be leading zeros in the string. The string is not in ASCII; to convert it to printable format, add the ASCII codes for ‘0’ or ‘A’, depending on the base and range.

The most significant limb of the input {s1p, s1n} must be non-zero. The input {s1p, s1n} is clobbered, except when base is a power of 2, in which case it’s unchanged.

The area at str has to have space for the largest possible number represented by a s1n long limb array, plus one extra character.

Function mp_size_t mpn set str (mp_limb_t *rp, const char *str, size_t strsize,

int base) Convert bytes {str,strsize} in the given base to limbs at rp.

str[0] is the most significant byte and str[strsize − 1] is the least significant. Each byte should be a value in the range 0 to base − 1, not an ASCII character. base can vary from 2 to 256.

The return value is the number of limbs written to rp. If the most significant input byte is non-zero then the high limb at rp will be non-zero, and only that exact number of limbs will be required there.

If the most significant input byte is zero then there may be high zero limbs written to rp and included in the return value.

strsize must be at least 1, and no overlap is permitted between {str,strsize} and the result at rp.

Function unsigned long int mpn scan0 (const mp_limb_t *s1p, unsigned long

int bit) Scan s1p from bit position bit for the next clear bit.

It is required that there be a clear bit within the area at s1p at or beyond bit position bit, so that the function has something to return.

Function unsigned long int mpn scan1 (const mp_limb_t *s1p, unsigned long

int bit) Scan s1p from bit position bit for the next set bit.

It is required that there be a set bit within the area at s1p at or beyond bit position bit, so that the function has something to return.

Function void mpn random (mp_limb_t *r1p, mp_size_t r1n) Function void mpn random2 (mp_limb_t *r1p, mp_size_t r1n)

Generate a random number of length r1n and store it at r1p. The most significant limb is always non-zero. mpn_random generates uniformly distributed limb data, mpn_random2 generates long strings of zeros and ones in the binary representation.

mpn_random2 is intended for testing the correctness of the mpn routines.

Function unsigned long int mpn popcount (const mp_limb_t *s1p, mp_size_t

n) Count the number of set bits in {s1p, n}.

Function unsigned long int mpn hamdist (const mp_limb_t *s1p, const
         mp_limb_t *s2p, mp_size_t n)

Compute the hamming distance between {s1p, n} and {s2p, n}.

Function int mpn perfect square p (const mp_limb_t *s1p, mp_size_t n)

Return non-zero iff {s1p, n} is a perfect square.

8.1 Nails

Everything in this section is highly experimental and may disappear or be subject to incompatible changes in a future version of GMP.

Nails are an experimental feature whereby a few bits are left unused at the top of each mp_limb_ t. This can significantly improve carry handling on some processors.

All the mpn functions accepting limb data will expect the nail bits to be zero on entry, and will return data with the nails similarly all zero. This applies both to limb vectors and to single limb arguments.

Nails can be enabled by configuring with ‘--enable-nails’. By default the number of bits will be chosen according to what suits the host processor, but a particular number can be selected

with ‘--enable-nails=N’.

At the mpn level, a nail build is neither source nor binary compatible with a non-nail build, strictly speaking. But programs acting on limbs only through the mpn functions are likely to work equally well with either build, and judicious use of the definitions below should make any program compatible with either build, at the source level.

For the higher level routines, meaning mpz etc, a nail build should be fully source and binary compatible with a non-nail build.

Macro GMP NAIL BITS Macro GMP NUMB BITS Macro GMP LIMB BITS

GMP_NAIL_BITS is the number of nail bits, or 0 when nails are not in use. GMP_NUMB_BITS is the number of data bits in a limb. GMP_LIMB_BITS is the total number of bits in an mp_limb_t. In all cases

          GMP_LIMB_BITS == GMP_NAIL_BITS + GMP_NUMB_BITS

Macro GMP NAIL MASK Macro GMP NUMB MASK

Bit masks for the nail and number parts of a limb. GMP_NAIL_MASK is 0 when nails are not in use.

GMP_NAIL_MASK is not often needed, since the nail part can be obtained with x >> GMP_ NUMB_BITS, and that means one less large constant, which can help various RISC chips.

Macro GMP NUMB MAX

The maximum value that can be stored in the number part of a limb. This is the same as GMP_NUMB_MASK, but can be used for clarity when doing comparisons rather than bit-wise operations.

The term “nails” comes from finger or toe nails, which are at the ends of a limb (arm or leg). “numb” is short for number, but is also how the developers felt after trying for a long time to come up with sensible names for these things.

In the future (the distant future most likely) a non-zero nail might be permitted, giving nonunique representations for numbers in a limb vector. This would help vector processors since carries would only ever need to propagate one or two limbs.

9 Random Number Functions

Sequences of pseudo-random numbers in GMP are generated using a variable of type gmp_ randstate_t, which holds an algorithm selection and a current state. Such a variable must be initialized by a call to one of the gmp_randinit functions, and can be seeded with one of the

gmp_randseed functions.

The functions actually generating random numbers are described in Section 5.13 [Integer Random Numbers], page 36, and Section 7.8 [Miscellaneous Float Functions], page 49.

The older style random number functions don’t accept a gmp_randstate_t parameter but instead share a global variable of that type. They use a default algorithm and are currently not seeded (though perhaps that will change in the future). The new functions accepting a gmp_randstate_t are recommended for applications that care about randomness.

9.1 Random State Initialization

Function void gmp randinit default (gmp_randstate_t state)

Initialize state with a default algorithm. This will be a compromise between speed and randomness, and is recommended for applications with no special requirements.

Function void gmp randinit lc 2exp (gmp_randstate_t state, mpz_t a,

    unsigned long c, unsigned long m2exp)

Initialize state with a linear congruential algorithm X = (aX + c) mod 2^{m2exp}.

The low bits of X in this algorithm are not very random. The least significant bit will have a period no more than 2, and the second bit no more than 4, etc. For this reason only the high half of each X is actually used.

When a random number of more than m2exp/2 bits is to be generated, multiple iterations of the recurrence are used and the results concatenated.

Function int gmp randinit lc 2exp size (gmp_randstate_t state, unsigned long

size) Initialize state for a linear congruential algorithm as per gmp_randinit_lc_2exp. a, c and m2exp are selected from a table, chosen so that size bits (or more) of each X will be used, ie. m2exp ≥ size/2.

If successful the return value is non-zero. If size is bigger than the table data provides then the return value is zero. The maximum size currently supported is 128.

Function void gmp randinit (gmp_randstate_t state, gmp_randalg_t alg, ...)

This function is obsolete.

Initialize state with an algorithm selected by alg. The only choice is GMP_RAND_ALG_

LC, which is gmp_randinit_lc_2exp_size. A third parameter of type unsigned long is

required, this is the size for that function. GMP_RAND_ALG_DEFAULT or 0 are the same as

GMP_RAND_ALG_LC.
gmp_randinit sets bits in gmp_errno to indicate an error. GMP_ERROR_UNSUPPORTED_

ARGUMENT if alg is unsupported, or GMP_ERROR_INVALID_ARGUMENT if the size parameter is too big.

Function void gmp randclear (gmp_randstate_t state)

Free all memory occupied by state.

9.2 Random State Seeding

Function void gmp randseed (gmp_randstate_t state, mpz_t seed) Function void gmp randseed ui (gmp_randstate_t state,

    unsigned long int seed)

Set an initial seed value into state.

The size of a seed determines how many different sequences of random numbers that it’s possible to generate. The “quality” of the seed is the randomness of a given seed compared to the previous seed used, and this affects the randomness of separate number sequences. The method for choosing a seed is critical if the generated numbers are to be used for important applications, such as generating cryptographic keys.

Traditionally the system time has been used to seed, but care needs to be taken with this. If an application seeds often and the resolution of the system clock is low, then the same sequence of numbers might be repeated. Also, the system time is quite easy to guess, so if unpredictability is required then it should definitely not be the only source for the seed value. On some systems there’s a special device ‘/dev/random’ which provides random data better suited for use as a seed.

10 Formatted Output

10.1 Format Strings

gmp_printf and friends accept format strings similar to the standard C printf (see section “Formatted Output” in The GNU C Library Reference Manual). A format specification is of the form

% [flags] [width] [.[precision]] [type] conv

GMP adds types ‘Z’, ‘Q’ and ‘F’ for mpz_t, mpq_t and mpf_t respectively, and ‘N’ for an mp_ limb_t array. ‘Z’, ‘Q’ and ‘N’ behave like integers. ‘Q’ will print a ‘/’ and a denominator, if needed. ‘F’ behaves like a float. For example,

mpz_t z;
gmp_printf ("%s is an mpz %Zd\n", "here", z);
mpq_t q;
gmp_printf ("a hex rational: %#40Qx\n", q);
mpf_t f;
int   n;
gmp_printf ("fixed point mpf %.*Ff with %d digits\n", n, f, n);
const mp_limb_t *ptr;
mp_size_t       size;
gmp_printf ("limb array %Nx\n", ptr, size);

For ‘N’ the limbs are expected least significant first, as per the mpn functions (see Chapter 8 [Low-level Functions], page 50). A negative size can be given to print the value as a negative.

All the standard C printf types behave the same as the C library printf, and can be freely intermixed with the GMP extensions. In the current implementation the standard parts of the format string are simply handed to printf and only the GMP extensions handled directly.

The flags accepted are as follows. GLIBC style ‘'’ is only for the standard C types (not the GMP types), and only if the C library supports it.

0  pad with zeros (rather than spaces)
#  show the base with ‘0x’, ‘0X’ or ‘0’
+  always show a sign
(space) show a space or a ‘-’ sign
'  group digits, GLIBC style (not GMP types)

The optional width and precision can be given as a number within the format string, or as a ‘*’ to take an extra parameter of type int, the same as the standard printf.

The standard types accepted are as follows. ‘h’ and ‘l’ are portable, the rest will depend on the compiler (or include files) for the type and the C library for the output.

h       short
hh      char
j       intmax_t or uintmax_t
l       long or wchar_t

ll  same as L

L       long long or long double
q       quad_t or u_quad_t
t       ptrdiff_t

     z       size_t

The GMP types are

F       mpf_t, float conversions
Q       mpq_t, integer conversions
N       mp_limb_t array, integer conversions
Z       mpz_t, integer conversions

The conversions accepted are as follows. ‘a’ and ‘A’ are always supported for mpf_t but depend on the C library for standard C float types. ‘m’ and ‘p’ depend on the C library.

a A  hex floats, GLIBC style
c  character
d  decimal integer
e E  scientific format float
f  fixed point float
i  same as d
g G  fixed or scientific float
m       strerror string, GLIBC style
n  store characters written so far
o  octal integer
p  pointer
s  string
u  unsigned integer
x X  hex integer

‘o’, ‘x’ and ‘X’ are unsigned for the standard C types, but for types ‘Z’, ‘Q’ and ‘N’ they are signed. ‘u’ is not meaningful for ‘Z’, ‘Q’ and ‘N’.

‘n’ can be used with any type, even the GMP types.

Other types or conversions that might be accepted by the C library printf cannot be used through gmp_printf, this includes for instance extensions registered with GLIBC register_ printf_function. Also currently there’s no support for POSIX ‘$’ style numbered arguments (perhaps this will be added in the future).

The precision field has it’s usual meaning for integer ‘Z’ and float ‘F’ types, but is currently undefined for ‘Q’ and should not be used with that.

mpf_t conversions only ever generate as many digits as can be accurately represented by the operand, the same as mpf_get_str does. Zeros will be used if necessary to pad to the requested precision. This happens even for an ‘f’ conversion of an mpf_t which is an integer, for instance 2^{1024} in an mpf_t of 128 bits precision will only produce about 20 digits, then pad with zeros to the decimal point. An empty precision field like ‘%.Fe’ or ‘%.Ff’ can be used to specifically request just the significant digits.

The decimal point character (or string) is taken from the current locale settings on systems which provide localeconv (see section “Locales and Internationalization” in The GNU C Library Reference Manual). The C library will normally do the same for standard float output.

10.2 Functions

Each of the following functions is similar to the corresponding C library function. The basic printf forms take a variable argument list. The vprintf forms take an argument pointer, see section “Variadic Functions” in The GNU C Library Reference Manual, or ‘man 3 va_start’.

It should be emphasised that if a format string is invalid, or the arguments don’t match what the format specifies, then the behaviour of any of these functions will be unpredictable. GCC format string checking is not available, since it doesn’t recognise the GMP extensions.

The file based functions gmp_printf and gmp_fprintf will return −1 to indicate a write error. All the functions can return −1 if the C library printf variant in use returns −1, but this shouldn’t normally occur.

Function int gmp printf (const char *fmt, ...) Function int gmp vprintf (const char *fmt, va_list ap)

Print to the standard output stdout. Return the number of characters written, or −1 if an error occurred.

Function int gmp fprintf (FILE *fp, const char *fmt, ...) Function int gmp vfprintf (FILE *fp, const char *fmt, va_list ap)

Print to the stream fp. Return the number of characters written, or −1 if an error occurred.

Function int gmp sprintf (char *buf, const char *fmt, ...) Function int gmp vsprintf (char *buf, const char *fmt, va_list ap)

Form a null-terminated string in buf. Return the number of characters written, excluding the terminating null.

No overlap is permitted between the space at buf and the string fmt.

These functions are not recommended, since there’s no protection against exceeding the space available at buf.

Function int gmp snprintf (char *buf, size_t size, const char *fmt, ...)

Function int gmp vsnprintf (char *buf, size_t size, const char *fmt, va_list

ap) Form a null-terminated string in buf. No more than size bytes will be written. To get the full output, size must be enough for the string and null-terminator.

The return value is the total number of characters which ought to have been produced, excluding the terminating null. If retval ≥ size then the actual output has been truncated to the first size − 1 characters, and a null appended.

No overlap is permitted between the region {buf,size} and the fmt string.

Notice the return value is in ISO C99 snprintf style. This is so even if the C library vsnprintf is the older GLIBC 2.0.x style.

Function int gmp asprintf (char **pp, const char *fmt, ...) Function int gmp vasprintf (char *pp, const char *fmt, va_list ap)

Form a null-terminated string in a block of memory obtained from the current memory allocation function (see Chapter 14 [Custom Allocation], page 78). The block will be the size of the string and null-terminator. Put the address of the block in *pp. Return the number of characters produced, excluding the null-terminator.

Unlike the C library asprintf, gmp_asprintf doesn’t return −1 if there’s no more memory available, it lets the current allocation function handle that.

Function int gmp obstack printf (struct obstack *ob, const char *fmt, ...)

Function int gmp obstack vprintf (struct obstack *ob, const char *fmt,

    va_list ap)

Append to the current obstack object, in the same style as obstack_printf. Return the number of characters written. A null-terminator is not written.

fmt cannot be within the current obstack object, since the object might move as it grows.

These functions are available only when the C library provides the obstack feature, which probably means only on GNU systems, see section “Obstacks” in The GNU C Library Reference Manual.

10.3 C++ Formatted Output

The following functions are provided in ‘libgmpxx’, which is built if C++ support is enabled (see Section 2.1 [Build Options], page 3). Prototypes are available from <gmp.h>.

Function ostream& operator<< (ostream& stream, mpz_t op)

Print op to stream, using its ios formatting settings. ios::width is reset to 0 after output, the same as the standard ostream operator<< routines do.

In hex or octal, op is printed as a signed number, the same as for decimal. This is unlike the standard operator<< routines on int etc, which instead give twos complement.

Function ostream& operator<< (ostream& stream, mpq_t op)

Print op to stream, using its ios formatting settings. ios::width is reset to 0 after output, the same as the standard ostream operator<< routines do.

Output will be a fraction like ‘5/9’, or if the denominator is 1 then just a plain integer like ‘123’.

In hex or octal, op is printed as a signed value, the same as for decimal. If ios::showbase is set then a base indicator is shown on both the numerator and denominator (if the denominator is required).

Function ostream& operator<< (ostream& stream, mpf_t op)

Print op to stream, using its ios formatting settings. ios::width is reset to 0 after output, the same as the standard ostream operator<< routines do. The decimal point follows the current locale, on systems providing localeconv.

Hex and octal are supported, unlike the standard operator<< on double. The mantissa will be in hex or octal, the exponent will be in decimal. For hex the exponent delimiter is an ‘@’. This is as per mpf_out_str.

ios::showbase is supported, and will put a base on the mantissa, for example hex ‘0x1.8’ or ‘0x0.8’, or octal ‘01.4’ or ‘00.4’. This last form is slightly strange, but at least differentiates itself from decimal.

These operators mean that GMP types can be printed in the usual C++ way, for example,

mpz_t  z;
int    n;
...
cout << "iteration " << n << " value " << z << "\n";

But note that ostream output (and istream input, see Section 11.3 [C++ Formatted Input], page 67) is the only overloading available and using for instance + with an mpz_t will have unpredictable results.

11 Formatted Input

11.1 Formatted Input Strings

gmp_scanf and friends accept format strings similar to the standard C scanf (see section “Formatted Input” in The GNU C Library Reference Manual). A format specification is of the form

% [flags] [width] [type] conv

GMP adds types ‘Z’, ‘Q’ and ‘F’ for mpz_t, mpq_t and mpf_t respectively. ‘Z’ and ‘Q’ behave like integers. ‘Q’ will read a ‘/’ and a denominator, if present. ‘F’ behaves like a float.

GMP variables don’t require an & when passed to gmp_scanf, since they’re already “call-by-reference”. For example,

/* to read say "a(5) = 1234" */
int   n;
mpz_t z;
gmp_scanf ("a(%d) = %Zd\n", &n, z);
mpq_t q1, q2;
gmp_sscanf ("0377 + 0x10/0x11", "%Qi + %Qi", q1, q2);
/* to read say "topleft (1.55,-2.66)" */
mpf_t x, y;
char  buf[32];
gmp_scanf ("%31s (%Ff,%Ff)", buf, x, y);

All the standard C scanf types behave the same as in the C library scanf, and can be freely intermixed with the GMP extensions. In the current implementation the standard parts of the format string are simply handed to scanf and only the GMP extensions handled directly.

The flags accepted are as follows. ‘a’ and ‘'’ will depend on support from the C library, and ‘'’ cannot be used with GMP types.

*  read but don’t store
a  allocate a buffer (string conversions)
'  group digits, GLIBC style (not GMP types)

The standard types accepted are as follows. ‘h’ and ‘l’ are portable, the rest will depend on the compiler (or include files) for the type and the C library for the input.

h       short
hh      char
j       intmax_t or uintmax_t
l       long or wchar_t

ll  same as L

L       long long or long double
q       quad_t or u_quad_t
t       ptrdiff_t
z       size_t

The GMP types are

F       mpf_t, float conversions
Q       mpq_t, integer conversions
Z       mpz_t, integer conversions

The conversions accepted are as follows. ‘p’ and ‘[’ will depend on support from the C library, the rest are standard.

c  character or characters
d  decimal integer

     {e E f g \atop G} float

i  integer with base indicator
n  characters written so far
o  octal integer
p  pointer
s  string of non-whitespace characters
u  decimal integer
x X  hex integer
[  string of characters in a set

‘e’, ‘E’, ‘f’, ‘g’ and ‘G’ are identical, they all read either fixed point or scientific format, and either ‘e’ or ‘E’ for the exponent in scientific format.

‘x’ and ‘X’ are identical, both accept both upper and lower case hexadecimal.

‘o’, ‘u’, ‘x’ and ‘X’ all read positive or negative values. For the standard C types these are described as “unsigned” conversions, but that merely affects certain overflow handling, negatives are still allowed (see strtoul, section “Parsing of Integers” in The GNU C Library Reference Manual). For GMP types there are no overflows, and ‘d’ and ‘u’ are identical.

‘Q’ type reads the numerator and (optional) denominator as given. If the value might not be in canonical form then mpq_canonicalize must be called before using it in any calculations (see Chapter 6 [Rational Number Functions], page 39).

‘Qi’ will read a base specification separately for the numerator and denominator. For example ‘0x10/11’ would be 16/11, whereas ‘0x10/0x11’ would be 16/17.

‘n’ can be used with any of the types above, even the GMP types. ‘*’ to suppress assignment is allowed, though the field would then do nothing at all.

Other conversions or types that might be accepted by the C library scanf cannot be used

through gmp_scanf.

Whitespace is read and discarded before a field, except for ‘c’ and ‘[’ conversions.

For float conversions, the decimal point character (or string) expected is taken from the current locale settings on systems which provide localeconv (see section “Locales and Internationalization” in The GNU C Library Reference Manual). The C library will normally do the same for standard float input.

11.2 Formatted Input Functions

Each of the following functions is similar to the corresponding C library function. The plain scanf forms take a variable argument list. The vscanf forms take an argument pointer, see section “Variadic Functions” in The GNU C Library Reference Manual, or ‘man 3 va_start’.

It should be emphasised that if a format string is invalid, or the arguments don’t match what the format specifies, then the behaviour of any of these functions will be unpredictable. GCC format string checking is not available, since it doesn’t recognise the GMP extensions.

No overlap is permitted between the fmt string and any of the results produced.

Function int gmp scanf (const char *fmt, ...) Function int gmp vscanf (const char *fmt, va_list ap)

Read from the standard input stdin.

Function int gmp fscanf (FILE *fp, const char *fmt, ...)

Function int gmp vfscanf (FILE *fp, const char *fmt, va_list ap)

Read from the stream fp.

Function int gmp sscanf (const char *s, const char *fmt, ...)
Function int gmp vsscanf (const char *s, const char *fmt, va_list ap)

Read from a null-terminated string s.

The return value from each of these functions is the same as the standard C99 scanf, namely the number of fields successfully parsed and stored. ‘%n’ fields and fields read but suppressed by ‘*’ don’t count towards the return value.

If end of file or file error, or end of string, is reached when a match is required, and when no previous non-suppressed fields have matched, then the return value is EOF instead of 0. A match is required for a literal character in the format string or a field other than ‘%n’. Whitespace in the format string is only an optional match and won’t induce an EOF in this fashion. Leading whitespace read and discarded for a field doesn’t count as a match.

11.3 C++ Formatted Input

The following functions are provided in ‘libgmpxx’, which is built only if C++ support is enabled (see Section 2.1 [Build Options], page 3). Prototypes are available from <gmp.h>.

Function istream& operator>> (istream& stream, mpz_t rop)

Read rop from stream, using its ios formatting settings.

Function istream& operator>> (istream& stream, mpq_t rop)

Read rop from stream, using its ios formatting settings.

An integer like ‘123’ will be read, or a fraction like ‘5/9’. If the fraction is not in canonical form then mpq_canonicalize must be called (see Chapter 6 [Rational Number Functions], page 39).

Function istream& operator>> (istream& stream, mpf_t rop)

Read rop from stream, using its ios formatting settings.

Hex or octal floats are not supported, but might be in the future.

These operators mean that GMP types can be read in the usual C++ way, for example,

     mpz_t  z;
     ...
     cin >> z;

But note that istream input (and ostream output, see Section 10.3 [C++ Formatted Output], page 63) is the only overloading available and using for instance + with an mpz_t will have unpredictable results.

12 C++ Class Interface

This chapter describes the C++ class based interface to GMP.

All GMP C language types and functions can be used in C++ programs, since ‘gmp.h’ has extern "C" qualifiers, but the class interface offers overloaded functions and operators which may be more convenient.

Due to the implementation of this interface, a reasonably recent C++ compiler is required, one supporting namespaces, partial specialization of templates and member templates. For GCC this means version 2.91 or later.

Everything described in this chapter is to be considered preliminary and might be subject to incompatible changes if some unforeseen difficulty reveals itself.

12.1 C++ Interface General

All the C++ classes and functions are available with

     #include <gmpxx.h>

The classes defined are

Class mpz class Class mpq class Class mpf class

The standard operators and various standard functions are overloaded to allow arithmetic with these classes. For example,

     int
     main (void)
     {
       mpz_class a, b, c;
       a = 1234;
       b = "-5678";
       c = a+b;
       cout << "sum is " << c << "\n";
       cout << "absolute value is " << abs(c) << "\n";
       return 0;
     }

An important feature of the implementation is that an expression like a=b+c results in a single call to the corresponding mpz_add, without using a temporary for the b+c part. Expressions which by their nature imply intermediate values, like a=b*c+d*e, still use temporaries though.

The classes can be freely intermixed in expressions, as can the classes and the standard types long, unsigned long and double. Smaller types like int or float can also be intermixed, since C++ will promote them.

Note that bool is not accepted directly, but must be explicitly cast to an int first. This is because C++ will automatically convert any pointer to a bool, so if GMP accepted bool it would make all sorts of invalid class and pointer combinations compile but almost certainly not do anything sensible.

Conversions back from the classes to standard C++ types aren’t done automatically, instead member functions like get_si are provided (see the following sections for details).

Also there are no automatic conversions from the classes to the corresponding GMP C types, instead a reference to the underlying C object can be obtained with the following functions,

Function mpz_t mpz class::get mpz t () Function mpq_t mpq class::get mpq t () Function mpf_t mpf class::get mpf t ()

These can be used to call a C function which doesn’t have a C++ class interface. For example to set a to the GCD of b and c,

mpz_class a, b, c;
...
mpz_gcd (a.get_mpz_t(), b.get_mpz_t(), c.get_mpz_t());

In the other direction, a class can be initialized from the corresponding GMP C type, or assigned to if an explicit constructor is used. In both cases this makes a copy of the value, it doesn’t create any sort of association. For example,

mpz_t z;
// ... init and calculate z ...
mpz_class x(z);
mpz_class y;
y = mpz_class (z);

There are no namespace setups in ‘gmpxx.h’, all types and functions are simply put into the global namespace. This is what ‘gmp.h’ has done in the past, and continues to do for compatibility. The extras provided by ‘gmpxx.h’ follow GMP naming conventions and are unlikely to clash with anything.

12.2 C++ Interface Integers

Function void mpz class::mpz class (type n)

Construct an mpz_class. All the standard C++ types may be used, except long long and long double, and all the GMP C++ classes can be used. Any necessary conversion follows the corresponding C function, for example double follows mpz_set_d (see Section 5.2 [Assigning Integers], page 26).

Function void mpz class::mpz class (mpz_t z)

Construct an mpz_class from an mpz_t. The value in z is copied into the new mpz_class, there won’t be any permanent association between it and z.

Function void mpz class::mpz class (const char *s) Function void mpz class::mpz class (const char *s, int base) Function void mpz class::mpz class (const string& s)

Function void mpz class::mpz class (const string& s, int base)

Construct an mpz_class converted from a string using mpz_set_str, (see Section 5.2 [Assigning Integers], page 26). If the base is not given then 0 is used.

Function mpz_class operator/ (mpz_class a, mpz_class d)
Function mpz_class operator% (mpz_class a, mpz_class d)

Divisions involving mpz_class round towards zero, as per the mpz_tdiv_q and mpz_tdiv_ r functions (see Section 5.6 [Integer Division], page 29). This corresponds to the rounding used for plain int calculations on most machines.

The mpz_fdiv... or mpz_cdiv... functions can always be called directly if desired. For example,

          mpz_class q, a, d;
          ...
          mpz_fdiv_q (q.get_mpz_t(), a.get_mpz_t(), d.get_mpz_t());

Function mpz_class abs (mpz_class op1)

Function int cmp (mpz_class op1, type op2) Function int cmp (type op1, mpz_class op2)

Function double mpz class::get d (void) Function long mpz class::get si (void) Function unsigned long mpz class::get ui (void) Function bool mpz class::fits sint p (void) Function bool mpz class::fits slong p (void) Function bool mpz class::fits sshort p (void) Function bool mpz class::fits uint p (void) Function bool mpz class::fits ulong p (void) Function bool mpz class::fits ushort p (void)

Function int sgn (mpz_class op)

Function mpz_class sqrt (mpz_class op)

These functions provide a C++ class interface to the corresponding GMP C routines.

cmp can be used with any of the classes or the standard C++ types, except long long and

     long double.

Overloaded operators for combinations of mpz_class and double are provided for completeness, but it should be noted that if the given double is not an integer then the way any rounding is done is currently unspecified. The rounding might take place at the start, in the middle, or at the end of the operation, and it might change in the future.

Conversions between mpz_class and double, however, are defined to follow the corresponding C functions mpz_get_d and mpz_set_d. And comparisons are always made exactly, as per

mpz_cmp_d.

12.3 C++ Interface Rationals

In all the following constructors, if a fraction is given then it should be in canonical form, or if

not then mpq_class::canonicalize called.

Function void mpq class::mpq class (type op)

Function void mpq class::mpq class (integer num, integer den)

Construct an mpq_class. The initial value can be a single value of any type, or a pair of integers (mpz_class or standard C++ integer types) representing a fraction, except that long long and long double are not supported. For example,

          mpq_class q (99);
          mpq_class q (1.75);
          mpq_class q (1, 3);

Function void mpq class::mpq class (mpq_t q)

Construct an mpq_class from an mpq_t. The value in q is copied into the new mpq_class, there won’t be any permanent association between it and q.

Function void mpq class::mpq class (const char *s) Function void mpq class::mpq class (const char *s, int base) Function void mpq class::mpq class (const string& s)

Function void mpq class::mpq class (const string& s, int base)

Construct an mpq_class converted from a string using mpq_set_str, (see Section 6.1 [Initializing Rationals], page 39). If the base is not given then 0 is used.

Function void mpq class::canonicalize ()

Put an mpq_class into canonical form, as per Chapter 6 [Rational Number Functions], page 39. All arithmetic operators require their operands in canonical form, and will return results in canonical form.

Function mpq_class abs (mpq_class op)

Function int cmp (mpq_class op1, type op2) Function int cmp (type op1, mpq_class op2)

Function double mpq class::get d (void)

Function int sgn (mpq_class op)

These functions provide a C++ class interface to the corresponding GMP C routines.

cmp can be used with any of the classes or the standard C++ types, except long long and

long double.

Function mpz_class& mpq class::get num () Function mpz_class& mpq class::get den ()

Get a reference to an mpz_class which is the numerator or denominator of an mpq_class. This can be used both for read and write access. If the object returned is modified, it modifies the original mpq_class.

If direct manipulation might produce a non-canonical value, then mpq_ class::canonicalize must be called before further operations.

Function mpz_t mpq class::get num mpz t () Function mpz_t mpq class::get den mpz t ()

Get a reference to the underlying mpz_t numerator or denominator of an mpq_class. This can be passed to C functions expecting an mpz_t. Any modifications made to the mpz_t will modify the original mpq_class.

If direct manipulation might produce a non-canonical value, then mpq_ class::canonicalize must be called before further operations.

Function istream& operator>> (istream& stream, mpq_class& rop);

Read rop from stream, using its ios formatting settings, the same as mpq_t operator>> (see Section 11.3 [C++ Formatted Input], page 67).

If the rop read might not be in canonical form then mpq_class::canonicalize must be called.

12.4 C++ Interface Floats

When an expression requires the use of temporary intermediate mpf_class values, like f=g*h+x*y, those temporaries will have the same precision as the destination f. Explicit constructors can be used if this doesn’t suit.

Function mpf class::mpf class (type op)

Function mpf class::mpf class (type op, unsigned long prec)

Construct an mpf_class. Any standard C++ type can be used, except long long and long double, and any of the GMP C++ classes can be used.

If prec is given, the initial precision is that value, in bits. If prec is not given, then the initial precision is determined by the type of op given. An mpz_class, mpq_class, string, or C++ builtin type will give the default mpf precision (see Section 7.1 [Initializing Floats], page 43). An mpf_class or expression will give the precision of that value. The precision of a binary expression is the higher of the two operands.

     mpf_class f(1.5);        // default precision
     mpf_class f(1.5, 500);   // 500 bits (at least)
     mpf_class f(x);          // precision of x
     mpf_class f(abs(x));     // precision of x
     mpf_class f(-g, 1000);   // 1000 bits (at least)
     mpf_class f(x+y);        // greater of precisions of x and y

Function mpf_class abs (mpf_class op)
Function mpf_class ceil (mpf_class op)

Function int cmp (mpf_class op1, type op2) Function int cmp (type op1, mpf_class op2)

Function mpf_class floor (mpf_class op)
Function mpf_class hypot (mpf_class op1, mpf_class op2)

Function double mpf class::get d (void) Function long mpf class::get si (void) Function unsigned long mpf class::get ui (void) Function bool mpf class::fits sint p (void) Function bool mpf class::fits slong p (void) Function bool mpf class::fits sshort p (void) Function bool mpf class::fits uint p (void) Function bool mpf class::fits ulong p (void) Function bool mpf class::fits ushort p (void)

Function int sgn (mpf_class op)

Function mpf_class sqrt (mpf_class op)
Function mpf_class trunc (mpf_class op)

These functions provide a C++ class interface to the corresponding GMP C routines.

cmp can be used with any of the classes or the standard C++ types, except long long and

long double.

The accuracy provided by hypot is not currently guaranteed.

Function unsigned long int mpf class::get prec () Function void mpf class::set prec (unsigned long prec) Function void mpf class::set prec raw (unsigned long prec)

Get or set the current precision of an mpf_class.

The restrictions described for mpf_set_prec_raw (see Section 7.1 [Initializing Floats], page 43) apply to mpf_class::set_prec_raw. Note in particular that the mpf_class must be restored to it’s allocated precision before being destroyed. This must be done by application code, there’s no automatic mechanism for it.

12.5 C++ Interface MPFR

The C++ class interface to MPFR is provided if MPFR is enabled (see Section 2.1 [Build Options], page 3). This interface must be regarded as preliminary and possibly subject to incompatible changes in the future, since MPFR itself is preliminary. All definitions can be obtained with

#include <mpfrxx.h>

This defines

Class mpfr class

which behaves similarly to mpf_class (see Section 12.4 [C++ Interface Floats], page 71).

12.6 C++ Interface Random Numbers

Class gmp randclass

The C++ class interface to the GMP random number functions uses gmp_randclass to hold an algorithm selection and current state, as per gmp_randstate_t.

Function gmp randclass::gmp randclass (void (*randinit) (gmp_randstate_t,

    ...), ...)

Construct a gmp_randclass, using a call to the given randinit function (see Section 9.1 [Random State Initialization], page 58). The arguments expected are the same as randinit, but with mpz_class instead of mpz_t. For example,

     gmp_randclass r1 (gmp_randinit_default);
     gmp_randclass r2 (gmp_randinit_lc_2exp_size, 32);
     gmp_randclass r3 (gmp_randinit_lc_2exp, a, c, m2exp);

gmp_randinit_lc_2exp_size can fail if the size requested is too big, the behaviour of gmp_randclass::gmp_randclass is undefined in this case (perhaps this will change in the future).

Function gmp randclass::gmp randclass (gmp_randalg_t alg, ...)

Construct a gmp_randclass using the same parameters as gmp_randinit (see Section 9.1 [Random State Initialization], page 58). This function is obsolete and the above randinit style should be preferred.

Function void gmp randclass::seed (unsigned long int s) Function void gmp randclass::seed (mpz_class s)

Seed a random number generator. See see Chapter 9 [Random Number Functions], page 58, for how to choose a good seed.

Function mpz_class gmp randclass::get z bits (unsigned long bits) Function mpz_class gmp randclass::get z bits (mpz_class bits)

Generate a random integer with a specified number of bits.

Function mpz_class gmp randclass::get z range (mpz_class n)

Generate a random integer in the range 0 to n − 1 inclusive.

Function mpf_class gmp randclass::get f () Function mpf_class gmp randclass::get f (unsigned long prec)

Generate a random float f in the range 0 <= f < 1. f will be to prec bits precision, or if prec is not given then to the precision of the destination. For example,

gmp_randclass  r;
...
mpf_class  f (0, 512);   // 512 bits precision
f = r.get_f();           // random number, 512 bits

12.7 C++ Interface Limitations

mpq_class and Templated Reading A generic piece of template code probably won’t know that mpq_class requires a canonicalize call if inputs read with operator>> might be non-canonical. This can lead to incorrect results. operator>> behaves as it does for reasons of efficiency. A canonicalize can be quite time consuming on large operands, and is best avoided if it’s not necessary. But this potential difficulty reduces the usefulness of mpq_class. Perhaps a mechanism to tell operator>> what to do will be adopted in the future, maybe a preprocessor define, a global flag, or an ios flag pressed into service. Or maybe, at the risk of inconsistency, the mpq_class operator>> could canonicalize and leave mpq_t operator>> not doing so, for use on those occasions when that’s acceptable. Send feedback or alternate ideas to bug-gmp@gnu.org.

Subclassing Subclassing the GMP C++ classes works, but is not currently recommended. Expressions involving subclasses resolve correctly (or seem to), but in normal C++ fashion the subclass doesn’t inherit constructors and assignments. There’s many of those in the GMP classes, and a good way to reestablish them in a subclass is not yet provided.

Templated Expressions A subtle difficulty exists when using expressions together with application-defined template functions. Consider the following, with T intended to be some numeric type,

     template <class T>
     T fun (const T &, const T &);

When used with, say, plain mpz_class variables, it works fine: T is resolved as

mpz_class.
     mpz_class f(1), g(2);
     fun (f, g);    // Good

But when one of the arguments is an expression, it doesn’t work.

     mpz_class f(1), g(2), h(3);
     fun (f, g+h);  // Bad

This is because g+h ends up being a certain expression template type internal to gmpxx.h, which the C++ template resolution rules are unable to automatically convert to mpz_class. The workaround is simply to add an explicit cast.

     mpz_class f(1), g(2), h(3);
     fun (f, mpz_class(g+h));  // Good

Similarly, within fun it may be necessary to cast an expression to type T when calling a templated fun2.

     template <class T>
     void fun (T f, T g)
     {
       fun2 (f, f+g);     // Bad
     }
     template <class T>

void fun (T f, T g)
{
  fun2 (f, T(f+g));  // Good
}

13 Berkeley MP Compatible Functions

These functions are intended to be fully compatible with the Berkeley MP library which is available on many BSD derived U*ix systems. The ‘--enable-mpbsd’ option must be used when building GNU MP to make these available (see Chapter 2 [Installing GMP], page 3).

The original Berkeley MP library has a usage restriction: you cannot use the same variable as both source and destination in a single function call. The compatible functions in GNU MP do not share this restriction—inputs and outputs may overlap.

It is not recommended that new programs are written using these functions. Apart from the incomplete set of functions, the interface for initializing MINT objects is more error prone, and the pow function collides with pow in ‘libm.a’.

Include the header ‘mp.h’ to get the definition of the necessary types and functions. If you are on a BSD derived system, make sure to include GNU ‘mp.h’ if you are going to link the GNU ‘libmp.a’ to your program. This means that you probably need to give the ‘-I<dir>’ option to the compiler, where ‘<dir>’ is the directory where you have GNU ‘mp.h’.

Function MINT * itom (signed short int initial value)

Allocate an integer consisting of a MINT object and dynamic limb space. Initialize the integer to initial value. Return a pointer to the MINT object.

Function MINT * xtom (char *initial value)

Allocate an integer consisting of a MINT object and dynamic limb space. Initialize the integer from initial value, a hexadecimal, null-terminated C string. Return a pointer to the MINT object.

Function void move (MINT *src, MINT *dest)

Set dest to src by copying. Both variables must be previously initialized.

Function void madd (MINT *src 1, MINT *src 2, MINT *destination)

Add src 1 and src 2 and put the sum in destination.

Function void msub (MINT *src 1, MINT *src 2, MINT *destination)

Subtract src 2 from src 1 and put the difference in destination.

Function void mult (MINT *src 1, MINT *src 2, MINT *destination)

Multiply src 1 and src 2 and put the product in destination.

Function void mdiv (MINT *dividend, MINT *divisor, MINT *quotient, MINT

*remainder)

Function void sdiv (MINT *dividend, signed short int divisor, MINT *quotient,

         signed short int *remainder)

Set quotient to dividend/divisor, and remainder to dividend mod divisor. The quotient is rounded towards zero; the remainder has the same sign as the dividend unless it is zero.

Some implementations of these functions work differently—or not at all—for negative arguments.

Function void msqrt (MINT *op, MINT *root, MINT *remainder)

Set root to ⌊\sqrt{op}⌋, like mpz_sqrt. Set remainder to (op −root^{2}), i.e. zero if op is a perfect square.

If root and remainder are the same variable, the results are undefined.

Function void pow (MINT *base, MINT *exp, MINT *mod, MINT *dest)

Set dest to (base raised to exp) modulo mod.

Function void rpow (MINT *base, signed short int exp, MINT *dest)

Set dest to base raised to exp.

Function void gcd (MINT *op1, MINT *op2, MINT *res)

Set res to the greatest common divisor of op1 and op2.

Function int mcmp (MINT *op1, MINT *op2)

Compare op1 and op2. Return a positive value if op1 > op2, zero if op1 = op2, and a negative value if op1 < op2.

Function void min (MINT *dest)

Input a decimal string from stdin, and put the read integer in dest. SPC and TAB are allowed in the number string, and are ignored.

Function void mout (MINT *src)

Output src to stdout, as a decimal string. Also output a newline.

Function char * mtox (MINT *op)

Convert op to a hexadecimal string, and return a pointer to the string. The returned string is allocated using the default memory allocation function, malloc by default. It will be strlen(str)+1 bytes, that being exactly enough for the string and null-terminator.

Function void mfree (MINT *op)

De-allocate, the space used by op. This function should only be passed a value returned

by itom or xtom.

14 Custom Allocation

By default GMP uses malloc, realloc and free for memory allocation, and if they fail GMP prints a message to the standard error output and terminates the program.

Alternate functions can be specified to allocate memory in a different way or to have a different error action on running out of memory.

This feature is available in the Berkeley compatibility library (see Chapter 13 [BSD Compatible Functions], page 76) as well as the main GMP library.

Function void mp set memory functions (

void *(*alloc func ptr) (size_t),

    void *(*realloc func ptr) (void *, size_t, size_t),
    void (*free func ptr) (void *, size_t))

Replace the current allocation functions from the arguments. If an argument is NULL, the corresponding default function is used.

These functions will be used for all memory allocation done by GMP, apart from temporary space from alloca if that function is available and GMP is configured to use it (see Section 2.1 [Build Options], page 3).

Be sure to call mp_set_memory_functions only when there are no active GMP objects allocated using the previous memory functions! Usually that means calling it before any other GMP function.

The functions supplied should fit the following declarations:

Function void * allocate function (size_t alloc size)

Return a pointer to newly allocated space with at least alloc size bytes.

Function void * reallocate function (void *ptr, size_t old size, size_t

new size) Resize a previously allocated block ptr of old size bytes to be new size bytes.

The block may be moved if necessary or if desired, and in that case the smaller of old size and new size bytes must be copied to the new location. The return value is a pointer to the resized block, that being the new location if moved or just ptr if not.

ptr is never NULL, it’s always a previously allocated block. new size may be bigger or smaller than old size.

Function void deallocate function (void *ptr, size_t size)

De-allocate the space pointed to by ptr.

ptr is never NULL, it’s always a previously allocated block of size bytes.

A byte here means the unit used by the sizeof operator.

The old size parameters to reallocate function and deallocate function are passed for convenience, but of course can be ignored if not needed. The default functions using malloc and friends for instance don’t use them.

No error return is allowed from any of these functions, if they return then they must have performed the specified operation. In particular note that allocate function or reallocate function mustn’t return NULL.

Getting a different fatal error action is a good use for custom allocation functions, for example giving a graphical dialog rather than the default print to stderr. How much is possible when genuinely out of memory is another question though.

There’s currently no defined way for the allocation functions to recover from an error such as out of memory, they must terminate program execution. A longjmp or throwing a C++ exception will have undefined results. This may change in the future.

GMP may use allocated blocks to hold pointers to other allocated blocks. This will limit the assumptions a conservative garbage collection scheme can make.

Since the default GMP allocation uses malloc and friends, those functions will be linked in even if the first thing a program does is an mp_set_memory_functions. It’s necessary to change the GMP sources if this is a problem.

15 Language Bindings

The following packages and projects offer access to GMP from languages other than C, though perhaps with varying levels of functionality and efficiency.

C++

• GMP C++ class interface, see Chapter 12 [C++ Class Interface], page 68 Straightforward interface, expression templates to eliminate temporaries.

• ALP http://www.inria.fr/saga/logiciels/ALP

Linear algebra and polynomials using templates.

• CLN http://clisp.cons.org/~haible/packages-cln.html

High level classes for arithmetic.

• LiDIA http://www.informatik.tu-darmstadt.de/TI/LiDIA

A C++ library for computational number theory.

• NTL http://www.shoup.net/ntl

A C++ number theory library.

Fortran

• Omni F77 http://pdplab.trc.rwcp.or.jp/pdperf/Omni/home.html

Arbitrary precision floats.

Haskell

• Glasgow Haskell Compiler http://www.haskell.org/ghc

Java

• Kaffe http://www.kaffe.org
• Kissme http://kissme.sourceforge.net

Lisp

• GNU Common Lisp http://www.gnu.org/software/gcl/gcl.html

In the process of switching to GMP for bignums.

• Librep http://librep.sourceforge.net

M4

• GNU m4 betas http://www.seindal.dk/rene/gnu

Optionally provides an arbitrary precision mpeval.

ML

• MLton compiler http://www.sourcelight.com/MLton

Oz

• Mozart http://www.mozart-oz.org

Pascal

• GNU Pascal Compiler http://www.gnu-pascal.de

GMP unit.

Perl • GMP module, see ‘demos/perl’ in the GMP sources.

• Math::GMP http://www.cpan.org

Compatible with Math::BigInt, but not as many functions as the GMP module above.

• Math::BigInt::GMP http://www.cpan.org

Plug Math::GMP into normal Math::BigInt operations.

Pike

• mpz module in the standard distribution, http://pike.idonex.com

Prolog

• SWI Prolog http://www.swi.psy.uva.nl/projects/SWI-Prolog

Arbitrary precision floats.

Python • mpz module in the standard distribution, http://www.python.org

• GMPY http://gmpy.sourceforge.net

Scheme

• RScheme http://www.rscheme.org
• STklos http://kaolin.unice.fr/STklos

Smalltalk

• GNU Smalltalk http://www.smalltalk.org/versions/GNUSmalltalk.html

Other

• DrGenius http://drgenius.seul.org

Geometry system and mathematical programming language.

• GiNaC http://www.ginac.de

C++ computer algebra using CLN.

• Maxima http://www.ma.utexas.edu/users/wfs/maxima.html

Macsyma computer algebra using GCL.

• Q http://www.musikwissenschaft.uni-mainz.de/~ag/q

Equational programming system.

• Regina http://regina.sourceforge.net

Topological calculator.

• Yacas http://www.xs4all.nl/~apinkus/yacas.html

Yet another computer algebra system.

16 Algorithms

This chapter is an introduction to some of the algorithms used for various GMP operations. The code is likely to be hard to understand without knowing something about the algorithms.

Some GMP internals are mentioned, but applications that expect to be compatible with future GMP releases should take care to use only the documented functions.

16.1 Multiplication

N×N limb multiplications and squares are done using one of four algorithms, as the size N increases.

Algorithm  Threshold
Basecase  (none)

     Karatsuba  MUL_KARATSUBA_THRESHOLD
     Toom-3  MUL_TOOM3_THRESHOLD
     FFT  MUL_FFT_THRESHOLD

Similarly for squaring, with the SQR thresholds. Note though that the FFT is only used if GMP is configured with ‘--enable-fft’, see Section 2.1 [Build Options], page 3.

N×M multiplications of operands with different sizes above MUL_KARATSUBA_THRESHOLD are currently done by splitting into M×M pieces. The Karatsuba and Toom-3 routines then operate only on equal size operands. This is not very efficient, and is slated for improvement in the future.

16.1.1 Basecase Multiplication

Basecase N×M multiplication is a straightforward rectangular set of cross-products, the same as long multiplication done by hand and for that reason sometimes known as the schoolbook or grammar school method. This is an O(NM) algorithm. See Knuth section 4.3.1 algorithm M (see Appendix B [References], page 109), and the ‘mpn/generic/mul_basecase.c’ code.

Assembler implementations of mpn_mul_basecase are essentially the same as the generic C code, but have all the usual assembler tricks and obscurities introduced for speed.

A square can be done in roughly half the time of a multiply, by using the fact that the cross products above and below the diagonal are the same. A triangle of products below the diagonal is formed, doubled (left shift by one bit), and then the products on the diagonal added. This can be seen in ‘mpn/generic/sqr_basecase.c’. Again the assembler implementations take essentially the same approach.

u0 u1 u2 u3 u4

u0 u1 u2 u3 u4

d d

d d

d

In practice squaring isn’t a full 2× faster than multiplying, it’s usually around 1.5×. Less than 1.5× probably indicates mpn_sqr_basecase wants improving on that CPU.

On some CPUs mpn_mul_basecase can be faster than the generic C mpn_sqr_basecase. SQR_ BASECASE_THRESHOLD is the size at which to use mpn_sqr_basecase, this will be zero if that routine should be used always.

16.1.2 Karatsuba Multiplication

The Karatsuba multiplication algorithm is described in Knuth section 4.3.3 part A, and various other textbooks. A brief description is given here.

The inputs x and y are treated as each split into two parts of equal length (or the most significant part one limb shorter if N is odd).

high low

x_{1} x_{0}
y_{1} y_{0}

Let b be the power of 2 where the split occurs, ie. if x_{0} is k limbs (y_{0} the same) then b = 2k∗mp bits per limb. With that x = x_{1}b + x_{0} and y = y_{1}b + y_{0}, and the following holds,

xy = (b^{2} + b)x_{1}y_{1} − b(x_{1} − x_{0})(y_{1} − y_{0}) + (b + 1)x_{0}y_{0}

This formula means doing only three multiplies of (N/2)×(N/2) limbs, whereas a basecase multiply of N×N limbs is equivalent to four multiplies of (N/2)×(N/2). The factors (b^{2} + b) etc represent the positions where the three products must be added.

high low

x_{1}y_{1} x_{0}y_{0}
+  x_{1}y_{1}
+  x_{0}y_{0}
− (x_{1} − x_{0})(y_{1} − y_{0})

The term (x_{1} − x_{0})(y_{1} − y_{0}) is best calculated as an absolute value, and the sign used to choose to add or subtract. Notice the sum high(x_{0}y_{0}) + low(x_{1}y_{1}) occurs twice, so it’s possible to do 5k limb additions, rather than 6k, but in GMP extra function call overheads outweigh the saving.

Squaring is similar to multiplying, but with x = y the formula reduces to an equivalent with three squares,

x^{2} = (b^{2} + b)x_{1}^{2} − b(x_{1} − x_{0})^{2} + (b + 1)x_{0}^{2}

The final result is accumulated from those three squares the same way as for the three multiplies above. The middle term (x_{1} − x_{0})^{2} is now always positive.

A similar formula for both multiplying and squaring can be constructed with a middle term (x_{1} + x_{0})(y_{1} + y_{0}). But those sums can exceed k limbs, leading to more carry handling and additions than the form above.

Karatsuba multiplication is asymptotically an O(N^{1.585}) algorithm, the exponent being log 3/ log 2, representing 3 multiplies each 1/2 the size of the inputs. This is a big improvement over the basecase multiply at O(N^{2}) and the advantage soon overcomes the extra additions Karatsuba performs.

MUL_KARATSUBA_THRESHOLD can be as little as 10 limbs. The SQR threshold is usually about twice the MUL. The basecase algorithm will take a time of the form M(N) = aN^{2} + bN + c and the Karatsuba algorithm K(N) = 3M(N/2) + dN + e. Clearly per-crossproduct speedups in the basecase code reduce a and decrease the threshold, but linear style speedups reducing b will actually increase the threshold. The latter can be seen for instance when adding an optimized mpn_sqr_diagonal to mpn_sqr_basecase. Of course all speedups reduce total time, and in that sense the algorithm thresholds are merely of academic interest.

16.1.3 Toom-Cook 3-Way Multiplication

The Karatsuba formula is the simplest case of a general approach to splitting inputs that leads to both Toom-Cook and FFT algorithms. A description of Toom-Cook can be found in Knuth section 4.3.3, with an example 3-way calculation after Theorem A. The 3-way form used in GMP is described here.

The operands are each considered split into 3 pieces of equal length (or the most significant part 1 or 2 limbs shorter than the others).

high low

x_{2}  x_{1}  x_{0}
y_{2}  y_{1}  y_{0}

These parts are treated as the coefficients of two polynomials

X(t) = x_{2}t^{2} + x_{1}t + x_{0} Y (t) = y_{2}t^{2} + y_{1}t + y_{0}

Again let b equal the power of 2 which is the size of the x_{0}, x_{1}, y_{0} and y_{1} pieces, ie. if they’re k limbs each then b = 2k∗mp bits per limb. With this x = X(b) and y = Y (b).

Let a polynomial W (t) = X(t)Y (t) and suppose its coefficients are

W (t) = w_{4}t^{4} + w_{3}t^{3} + w_{2}t^{2} + w_{1}t + w_{0}

The w_{i} are going to be determined, and when they are they’ll give the final result using w = W (b), since xy = X(b)Y (b). The coefficients will be roughly b^{2} each, and the final W (b) will be an addition like,

high low

w_{4}

w_{3}

w_{2}

w_{1}

w_{1}

The w_{i} coefficients could be formed by a simple set of cross products, like w_{4} = x_{2}y_{2}, w_{3} = x_{2}y_{1} + x_{1}y_{2}, w_{2} = x_{2}y_{0} + x_{1}y_{1} + x_{0}y_{2} etc, but this would need all nine x_{i}y_{j} for i, j = 0, 1, 2, and would be equivalent merely to a basecase multiply. Instead the following approach is used.

X(t) and Y (t) are evaluated and multiplied at 5 points, giving values of W (t) at those points. The points used can be chosen in various ways, but in GMP the following are used

Point  Value
t = 0  x_{0}y_{0}, which gives w_{0} immediately
t = 2  (4x_{2} + 2x_{1} + x_{0})(4y_{2} + 2y_{1} + y_{0})
t = 1  (x_{2} + x_{1} + x_{0})(y_{2} + y_{1} + y_{0})
t = \frac{1}{2}  (x_{2} + 2x_{1} + 4x_{0})(y_{2} + 2y_{1} + 4y_{0})
t = ∞  x_{2}y_{2}, which gives w_{4} immediately

At t = \frac{1}{2} the value calculated is actually 16X(\frac{1}{2})Y (\frac{1}{2}), giving a value for 16W (\frac{1}{2}), and this is always an integer. At t = ∞ the value is actually lim_{t→∞} \frac{X(t)Y (t)}{t^{4}}, but it’s much easier to think of as simply x_{2}y_{2} giving w_{4} immediately (much like x_{0}y_{0} at t = 0 gives w_{0} immediately).

Now each of the points substituted into W (t) = w_{4}t^{4} + ⋯ + w_{0} gives a linear combination of the w_{i} coefficients, and the value of those combinations has just been calculated.

W (0) =  w_{0}
16W (\frac{1}{2}) = w_{4} + 2w_{3} + 4w_{2} + 8w_{1} + 16w_{0}
W (1) = w_{4} + w_{3} + w_{2} + w_{1} + w_{0}
W (2) = 16w_{4} + 8w_{3} + 4w_{2} + 2w_{1} + w_{0}
W (∞) = w_{4}

This is a set of five equations in five unknowns, and some elementary linear algebra quickly isolates each w_{i}, by subtracting multiples of one equation from another.

In the code the set of five values W (0),…,W (∞) will represent those certain linear combinations. By adding or subtracting one from another as necessary, values which are each w_{i} alone are arrived at. This involves only a few subtractions of small multiples (some of which are powers of 2), and so is fast. A couple of divisions remain by powers of 2 and one division by 3 (or by 6 rather), and that last uses the special mpn_divexact_by3 (see Section 16.2.4 [Exact Division], page 89).

In the code the values w_{4}, w_{2} and w_{0} are formed in the destination with pointers E, C and A, and w_{3} and w_{1} in temporary space D and B are added to them. There are extra limbs tD, tC and tB at the high end of w_{3}, w_{2} and w_{1} which are handled separately. The final addition then is as follows.

high low

              E                   C                   A
                        D                  B

            tD        tC        tB

The conversion of W (t) values to the coefficients is interpolation. A polynomial of degree 4 like W (t) is uniquely determined by values known at 5 different points. The points can be chosen to make the linear equations come out with a convenient set of steps for isolating the w_{i}.

In ‘mpn/generic/mul_n.c’ the interpolate3 routine performs the interpolation. The opencoded one-pass version may be a bit hard to understand, the steps performed can be better seen in the USE_MORE_MPN version.

Squaring follows the same procedure as multiplication, but there’s only one X(t) and it’s evaluated at 5 points, and those values squared to give values of W (t). The interpolation is then identical, and in fact the same interpolate3 subroutine is used for both squaring and multiplying.

Toom-3 is asymptotically O(N^{1.465}), the exponent being log 5/ log 3, representing 5 recursive multiplies of 1/3 the original size. This is an improvement over Karatsuba at O(N^{1.585}), though Toom-Cook does more work in the evaluation and interpolation and so it only realizes its advantage above a certain size.

Near the crossover between Toom-3 and Karatsuba there’s generally a range of sizes where the difference between the two is small. MUL_TOOM3_THRESHOLD is a somewhat arbitrary point in that range and successive runs of the tune program can give different values due to small variations in measuring. A graph of time versus size for the two shows the effect, see ‘tune/README’.

At the fairly small sizes where the Toom-3 thresholds occur it’s worth remembering that the asymptotic behaviour for Karatsuba and Toom-3 can’t be expected to make accurate predictions, due of course to the big influence of all sorts of overheads, and the fact that only a few recursions of each are being performed. Even at large sizes there’s a good chance machine dependent effects like cache architecture will mean actual performance deviates from what might be predicted.

The formula given above for the Karatsuba algorithm has an equivalent for Toom-3 involving only five multiplies, but this would be complicated and unenlightening.

An alternate view of Toom-3 can be found in Zuras (see Appendix B [References], page 109), using a vector to represent the x and y splits and a matrix multiplication for the evaluation and interpolation stages. The matrix inverses are not meant to be actually used, and they have elements with values much greater than in fact arise in the interpolation steps. The diagram shown for the 3-way is attractive, but again doesn’t have to be implemented that way and for example with a bit of rearrangement just one division by 6 can be done.

16.1.4 FFT Multiplication

At large to very large sizes a Fermat style FFT multiplication is used, following Schönhage and Strassen (see Appendix B [References], page 109). Descriptions of FFTs in various forms can be found in many textbooks, for instance Knuth section 4.3.3 part C or Lipson chapter IX. A brief description of the form used in GMP is given here.

The multiplication done is xy mod 2^{N} + 1, for a given N. A full product xy is obtained by choosing N ≥ bits(x)+bits(y) and padding x and y with high zero limbs. The modular product is the native form for the algorithm, so padding to get a full product is unavoidable.

The algorithm follows a split, evaluate, pointwise multiply, interpolate and combine similar to that described above for Karatsuba and Toom-3. A k parameter controls the split, with an FFTk splitting into 2^{k} pieces of M = N/2^{k} bits each. N must be a multiple of 2^{k} ×mp bits per limb so the split falls on limb boundaries, avoiding bit shifts in the split and combine stages.

The evaluations, pointwise multiplications, and interpolation, are all done modulo 2^{N′} +1 where N′ is 2M + k + 3 rounded up to a multiple of 2^{k} and of mp_bits_per_limb. The results of interpolation will be the following negacyclic convolution of the input pieces, and the choice of N′ ensures these sums aren’t truncated.

w_{n} = ∑_{{i+j=b2^{k}+n \atop b=0,1}}(−1)^{b}x_{i}y_{j}

The points used for the evaluation are g^{i} for i = 0 to 2^{k} − 1 where g = 2^{2N′/2^{k}}. g is a 2^{k}th root of unity mod 2^{N′} + 1, which produces necessary cancellations at the interpolation stage, and it’s also a power of 2 so the fast fourier transforms used for the evaluation and interpolation do only shifts, adds and negations.

The pointwise multiplications are done modulo 2^{N′} + 1 and either recurse into a further FFT or use a plain multiplication (Toom-3, Karatsuba or basecase), whichever is optimal at the size N′. The interpolation is an inverse fast fourier transform. The resulting set of sums of x_{i}y_{j} are added at appropriate offsets to give the final result.

Squaring is the same, but x is the only input so it’s one transform at the evaluate stage and the pointwise multiplies are squares. The interpolation is the same.

For a mod 2^{N} + 1 product, an FFT-k is an O(N^{k/(k−1)}) algorithm, the exponent representing 2^{k} recursed modular multiplies each 1/2^{k−1} the size of the original. Each successive k is an asymptotic improvement, but overheads mean each is only faster at bigger and bigger sizes. In the code, MUL_FFT_TABLE and SQR_FFT_TABLE are the thresholds where each k is used. Each new k effectively swaps some multiplying for some shifts, adds and overheads.

A mod 2^{N} +1 product can be formed with a normal N ×N → 2N bit multiply plus a subtraction, so an FFT and Toom-3 etc can be compared directly. A k = 4 FFT at O(N^{1.333}) can be expected to be the first faster than Toom-3 at O(N^{1.465}). In practice this is what’s found, with MUL_FFT_ MODF_THRESHOLD and SQR_FFT_MODF_THRESHOLD being between 300 and 1000 limbs, depending on the CPU. So far it’s been found that only very large FFTs recurse into pointwise multiplies above these sizes.

When an FFT is to give a full product, the change of N to 2N doesn’t alter the theoretical complexity for a given k, but for the purposes of considering where an FFT might be first used it can be assumed that the FFT is recursing into a normal multiply and that on that basis it’s doing 2^{k} recursed multiplies each 1/2^{k−2} the size of the inputs, making it O(N^{k/(k−2)}). This would mean k = 7 at O(N^{1.4}) would be the first FFT faster than Toom-3. In practice MUL_ FFT_THRESHOLD and SQR_FFT_THRESHOLD have been found to be in the k = 8 range, somewhere between 3000 and 10000 limbs.

The way N is split into 2^{k} pieces and then 2M + k + 3 is rounded up to a multiple of 2^{k} and mp_bits_per_limb means that when 2^{k} ≥ mp bits per limb the effective N is a multiple of 2^{2k−1} bits. The +k + 3 means some values of N just under such a multiple will be rounded to the next. The complexity calculations above assume that a favourable size is used, meaning one which isn’t padded through rounding, and it’s also assumed that the extra +k + 3 bits are negligible at typical FFT sizes.

The practical effect of the 2^{2k−1} constraint is to introduce a step-effect into measured speeds. For example k = 8 will round N up to a multiple of 32768 bits, so for a 32-bit limb there’ll be 512 limb groups of sizes for which mpn_mul_n runs at the same speed. Or for k = 9 groups of 2048 limbs, k = 10 groups of 8192 limbs, etc. In practice it’s been found each k is used at quite small multiples of its size constraint and so the step effect is quite noticeable in a time versus size graph.

The threshold determinations currently measure at the mid-points of size steps, but this is suboptimal since at the start of a new step it can happen that it’s better to go back to the previous k for a while. Something more sophisticated for MUL_FFT_TABLE and SQR_FFT_TABLE will be needed.

16.1.5 Other Multiplication

The 3-way Toom-Cook algorithm described above (see Section 16.1.3 [Toom-Cook 3-Way Multiplication], page 84) generalizes to split into an arbitrary number of pieces, as per Knuth section 4.3.3 algorithm C. This is not currently used, though it’s possible a Toom-4 might fit in between Toom-3 and the FFTs. The notes here are merely for interest.

In general a split into r + 1 pieces is made, and evaluations and pointwise multiplications done at 2r +1 points. A 4-way split does 7 pointwise multiplies, 5-way does 9, etc. Asymptotically an (r + 1)-way algorithm is O(N^{log(2r+1)/log(r+1)}. Only the pointwise multiplications count towards big-O complexity, but the time spent in the evaluate and interpolate stages grows with r and has a significant practical impact, with the asymptotic advantage of each r realized only at bigger and bigger sizes. The overheads grow as O(Nr), whereas in an r = 2^{k} FFT they grow only as O(N log r).

Knuth algorithm C evaluates at points 0,1,2,…,2r, but exercise 4 uses −r,…,0,…,r and the latter saves some small multiplies in the evaluate stage (or rather trades them for additions), and has a further saving of nearly half the interpolate steps. The idea is to separate odd and even final coefficients and then perform algorithm C steps C7 and C8 on them separately. The divisors at step C7 become j^{2} and the multipliers at C8 become 2tj − j^{2}.

Splitting odd and even parts through positive and negative points can be thought of as using −1 as a square root of unity. If a 4th root of unity was available then a further split and speedup would be possible, but no such root exists for plain integers. Going to complex integers with i = \sqrt{−1} doesn’t help, essentially because in cartesian form it takes three real multiplies to do a complex multiply. The existence of 2^{k}th roots of unity in a suitable ring or field lets the fast fourier transform keep splitting and get to O(N log r).

Floating point FFTs use complex numbers approximating Nth roots of unity. Some processors have special support for such FFTs. But these are not used in GMP since it’s very difficult to guarantee an exact result (to some number of bits). An occasional difference of 1 in the last bit might not matter to a typical signal processing algorithm, but is of course of vital importance to GMP.

16.2 Division Algorithms

16.2.1 Single Limb Division

N×1 division is implemented using repeated 2×1 divisions from high to low, either with a hardware divide instruction or a multiplication by inverse, whichever is best on a given CPU.

The multiply by inverse follows section 8 of “Division by Invariant Integers using Multiplication” by Granlund and Montgomery (see Appendix B [References], page 109) and is implemented as udiv_qrnnd_preinv in ‘gmp-impl.h’. The idea is to have a fixed-point approximation to 1/d (see invert_limb) and then multiply by the high limb (plus one bit) of the dividend to get a quotient q. With d normalized (high bit set), q is no more than 1 too small. Subtracting qd from the dividend gives a remainder, and reveals whether q or q − 1 is correct.

The result is a division done with two multiplications and four or five arithmetic operations. On CPUs with low latency multipliers this can be much faster than a hardware divide, though the cost of calculating the inverse at the start may mean it’s only better on inputs bigger than say 4 or 5 limbs.

When a divisor must be normalized, either for the generic C __udiv_qrnnd_c or the multiply by inverse, the division performed is actually a2^{k} by d2^{k} where a is the dividend and k is the power necessary to have the high bit of d2^{k} set. The bit shifts for the dividend are usually accomplished “on the fly” meaning by extracting the appropriate bits at each step. Done this way the quotient limbs come out aligned ready to store. When only the remainder is wanted, an alternative is to take the dividend limbs unshifted and calculate r = a mod d2^{k} followed by an extra final step r2^{k} mod d2^{k}. This can help on CPUs with poor bit shifts or few registers.

The multiply by inverse can be done two limbs at a time. The calculation is basically the same, but the inverse is two limbs and the divisor treated as if padded with a low zero limb. This means more work, since the inverse will need a 2×2 multiply, but the four 1×1s to do that are independent and can therefore be done partly or wholly in parallel. Likewise for a 2×1 calculating qd. The net effect is to process two limbs with roughly the same two multiplies worth of latency that one limb at a time gives. This extends to 3 or 4 limbs at a time, though the extra work to apply the inverse will almost certainly soon reach the limits of multiplier throughput.

A similar approach in reverse can be taken to process just half a limb at a time if the divisor is only a half limb. In this case the 1×1 multiply for the inverse effectively becomes two \frac{1}{2×1} for each limb, which can be a saving on CPUs with a fast half limb multiply, or in fact if the only multiply is a half limb, and especially if it’s not pipelined.

16.2.2 Basecase Division

Basecase N×M division is like long division done by hand, but in base 2mp bits per limb. See Knuth section 4.3.1 algorithm D, and ‘mpn/generic/sb_divrem_mn.c’.

Briefly stated, while the dividend remains larger than the divisor, a high quotient limb is formed and the N×1 product qd subtracted at the top end of the dividend. With a normalized divisor (most significant bit set), each quotient limb can be formed with a 2×1 division and a 1×1 multiplication plus some subtractions. The 2×1 division is by the high limb of the divisor and is done either with a hardware divide or a multiply by inverse (the same as in Section 16.2.1 [Single Limb Division], page 88) whichever is faster. Such a quotient is sometimes one too big, requiring an addback of the divisor, but that happens rarely.

With Q=N−M being the number of quotient limbs, this is an O(QM) algorithm and will run at a speed similar to a basecase Q×M multiplication, differing in fact only in the extra multiply and divide for each of the Q quotient limbs.

16.2.3 Divide and Conquer Division

For divisors larger than DIV_DC_THRESHOLD, division is done by dividing. Or to be precise by a recursive divide and conquer algorithm based on work by Moenck and Borodin, Jebelean, and Burnikel and Ziegler (see Appendix B [References], page 109).

The algorithm consists essentially of recognising that a 2N×N division can be done with the basecase division algorithm (see Section 16.2.2 [Basecase Division], page 88), but using N/2 limbs as a base, not just a single limb. This way the multiplications that arise are (N/2)×(N/2) and can take advantage of Karatsuba and higher multiplication algorithms (see Section 16.1 [Multiplication Algorithms], page 82). The “digits” of the quotient are formed by recursive N×(N/2) divisions.

If the (N/2)×(N/2) multiplies are done with a basecase multiplication then the work is about the same as a basecase division, but with more function call overheads and with some subtractions separated from the multiplies. These overheads mean that it’s only when N/2 is above MUL_ KARATSUBA_THRESHOLD that divide and conquer is of use.

DIV_DC_THRESHOLD is based on the divisor size N, so it will be somewhere above twice MUL_ KARATSUBA_THRESHOLD, but how much above depends on the CPU. An optimized mpn_mul_ basecase can lower DIV_DC_THRESHOLD a little by offering a ready-made advantage over repeated

mpn_submul_1 calls.

Divide and conquer is asymptotically O(M(N) log N) where M(N) is the time for an N×N multiplication done with FFTs. The actual time is a sum over multiplications of the recursed sizes, as can be seen near the end of section 2.2 of Burnikel and Ziegler. For example, within the Toom-3 range, divide and conquer is 2.63M(N). With higher algorithms the M(N) term improves and the multiplier tends to log N. In practice, at moderate to large sizes, a 2N×N division is about 2 to 4 times slower than an N×N multiplication.

Newton’s method used for division is asymptotically O(M(N)) and should therefore be superior to divide and conquer, but it’s believed this would only be for large to very large N.

16.2.4 Exact Division

A so-called exact division is when the dividend is known to be an exact multiple of the divisor. Jebelean’s exact division algorithm uses this knowledge to make some significant optimizations (see Appendix B [References], page 109).

The idea can be illustrated in decimal for example with 368154 divided by 543. Because the low digit of the dividend is 4, the low digit of the quotient must be 8. This is arrived at from 4×7 mod 10, using the fact 7 is the modular inverse of 3 (the low digit of the divisor), since 3×7 ≡ 1 mod 10. So 8×543 = 4344 can be subtracted from the dividend leaving 363810. Notice the low digit has become zero.

The procedure is repeated at the second digit, with the next quotient digit 7 (1×7 mod 10), subtracting 7×543 = 3801, leaving 325800. And finally at the third digit with quotient digit 6 (8×7 mod 10), subtracting 6×543 = 3258 leaving 0. So the quotient is 678.

Notice however that the multiplies and subtractions don’t need to extend past the low three digits of the dividend, since that’s enough to determine the three quotient digits. For the last quotient digit no subtraction is needed at all. On a 2N×N division like this one, only about half the work of a normal basecase division is necessary.

For an N×M exact division producing Q=N−M quotient limbs, the saving over a normal basecase division is in two parts. Firstly, each of the Q quotient limbs needs only one multiply, not a 2×1 divide and multiply. Secondly, the crossproducts are reduced when Q > M to QM −M(M +1)/2, or when Q ≤ M to Q(Q − 1)/2. Notice the savings are complementary. If Q is big then many divisions are saved, or if Q is small then the crossproducts reduce to a small number.

The modular inverse used is calculated efficiently by modlimb_invert in ‘gmp-impl.h’. This does four multiplies for a 32-bit limb, or six for a 64-bit limb. ‘tune/modlinv.c’ has some alternate implementations that might suit processors better at bit twiddling than multiplying.

The sub-quadratic exact division described by Jebelean in “Exact Division with Karatsuba Complexity” is not currently implemented. It uses a rearrangement similar to the divide and conquer for normal division (see Section 16.2.3 [Divide and Conquer Division], page 89), but operating from low to high. A further possibility not currently implemented is “Bidirectional Exact Integer Division” by Krandick and Jebelean which forms quotient limbs from both the high and low ends of the dividend, and can halve once more the number of crossproducts needed in a 2N×N division.

A special case exact division by 3 exists in mpn_divexact_by3, supporting Toom-3 multiplication and mpq canonicalizations. It forms quotient digits with a multiply by the modular inverse of 3 (which is 0xAA..AAB) and uses two comparisons to determine a borrow for the next limb. The multiplications don’t need to be on the dependent chain, as long as the effect of the borrows is applied. Only a few optimized assembler implementations currently exist.

16.2.5 Exact Remainder

If the exact division algorithm is done with a full subtraction at each stage and the dividend isn’t a multiple of the divisor, then low zero limbs are produced but with a remainder in the high limbs. For dividend a, divisor d, quotient q, and b = 2mp bits per limb, then this remainder r is of the form

a = qd + rb^{n}

n represents the number of zero limbs produced by the subtractions, that being the number of limbs produced for q. r will be in the range 0 ≤ r < d and can be viewed as a remainder, but one shifted up by a factor of b^{n}.

Carrying out full subtractions at each stage means the same number of cross products must be done as a normal division, but there’s still some single limb divisions saved. When d is a single limb some simplifications arise, providing good speedups on a number of processors.

mpn_bdivmod, mpn_divexact_by3, mpn_modexact_1_odd and the redc function in mpz_powm

differ subtly in how they return r, leading to some negations in the above formula, but all are essentially the same.

Clearly r is zero when a is a multiple of d, and this leads to divisibility or congruence tests which are potentially more efficient than a normal division.

The factor of b^{n} on r can be ignored in a GCD when d is odd, hence the use of mpn_bdivmod in

mpn_gcd, and the use of mpn_modexact_1_odd by mpn_gcd_1 and mpz_kronecker_ui etc (see

Section 16.3 [Greatest Common Divisor Algorithms], page 91).

Montgomery’s REDC method for modular multiplications uses operands of the form of xb^{−n} and yb^{−n} and on calculating (xb^{−n})(yb^{−n}) uses the factor of b^{n} in the exact remainder to reach a product in the same form (xy)b^{−n} (see Section 16.4.2 [Modular Powering Algorithm], page 93).

Notice that r generally gives no useful information about the ordinary remainder a mod d since b^{n} mod d could be anything. If however b^{n} ≡ 1 mod d, then r is the negative of the ordinary remainder. This occurs whenever d is a factor of b^{n} −1, as for example with 3 in mpn_divexact_ by3. Other such factors include 5, 17 and 257, but no particular use has been found for this.

16.2.6 Small Quotient Division

An N×M division where the number of quotient limbs Q=N−M is small can be optimized somewhat.

An ordinary basecase division normalizes the divisor by shifting it to make the high bit set, shifting the dividend accordingly, and shifting the remainder back down at the end of the calculation. This is wasteful if only a few quotient limbs are to be formed. Instead a division of just the top 2Q limbs of the dividend by the top Q limbs of the divisor can be used to form a trial quotient. This requires only those limbs normalized, not the whole of the divisor and dividend.

A multiply and subtract then applies the trial quotient to the M−Q unused limbs of the divisor and N−Q dividend limbs (which includes Q limbs remaining from the trial quotient division). The starting trial quotient can be 1 or 2 too big, but all cases of 2 too big and most cases of 1 too big are detected by first comparing the most significant limbs that will arise from the subtraction. An addback is done if the quotient still turns out to be 1 too big.

This whole procedure is essentially the same as one step of the basecase algorithm done in a Q limb base, though with the trial quotient test done only with the high limbs, not an entire Q limb “digit” product. The correctness of this weaker test can be established by following the argument of Knuth section 4.3.1 exercise 20 but with the v_{2}q̂ > br̂ + u_{2} condition appropriately relaxed.

16.3 Greatest Common Divisor

16.3.1 Binary GCD

At small sizes GMP uses an O(N^{2}) binary style GCD. This is described in many textbooks, for example Knuth section 4.5.2 algorithm B. It simply consists of successively reducing operands a and b using gcd (a, b) = gcd (min (a, b), abs (a − b)), and also that if a and b are first made odd then abs (a − b) is even and factors of two can be discarded.

Variants like letting a − b become negative and doing a different next step are of interest only as far as they suit particular CPUs, since on small operands it’s machine dependent factors that determine performance.

The Euclidean GCD algorithm, as per Knuth algorithms E and A, reduces using a mod b but this has so far been found to be slower everywhere. One reason the binary method does well is that the implied quotient at each step is usually small, so often only one or two subtractions are needed to get the same effect as a division. Quotients 1, 2 and 3 for example occur 67.7% of the time, see Knuth section 4.5.3 Theorem E.

When the implied quotient is large, meaning b is much smaller than a, then a division is worthwhile. This is the basis for the initial a mod b reductions in mpn_gcd and mpn_gcd_1 (the latter for both N×1 and 1×1 cases). But after that initial reduction, big quotients occur too rarely to make it worth checking for them.

16.3.2 Accelerated GCD

For sizes above GCD_ACCEL_THRESHOLD, GMP uses the Accelerated GCD algorithm described independently by Weber and Jebelean (the latter as the “Generalized Binary” algorithm), see Appendix B [References], page 109. This algorithm is still O(N^{2}), but is much faster than the binary algorithm since it does fewer multi-precision operations. It consists of alternating the k-ary reduction by Sorenson, and a “dmod” exact remainder reduction.

For operands u and v the k-ary reduction replaces u with nv − du where n and d are single limb values chosen to give two trailing zero limbs on that value, which can be stripped. n and d are calculated using an algorithm similar to half of a two limb GCD (see find_a in

‘mpn/generic/gcd.c’).

When u and v differ in size by more than a certain number of bits, a dmod is performed to zero out bits at the low end of the larger. It consists of an exact remainder style division applied to an appropriate number of bits (see Section 16.2.4 [Exact Division], page 89, and see Section 16.2.5 [Exact Remainder], page 90). This is faster than a k-ary reduction but useful only when the operands differ in size. There’s a dmod after each k-ary reduction, and if the dmod leaves the operands still differing in size then it’s repeated.

The k-ary reduction step can introduce spurious factors into the GCD calculated, and these are eliminated at the end by taking GCDs with the original inputs gcd (u, gcd (v, g)) using the binary algorithm. Since g is almost always small this takes very little time.

At small sizes the algorithm needs a good implementation of find_a. At larger sizes it’s dominated by mpn_addmul_1 applying n and d.

16.3.3 Extended GCD

The extended GCD calculates gcd (a, b) and also cofactors x and y satisfying ax+by = gcd(a,b). Lehmer’s multi-step improvement of the extended Euclidean algorithm is used. See Knuth section 4.5.2 algorithm L, and ‘mpn/generic/gcdext.c’. This is an O(N^{2}) algorithm.

The multipliers at each step are found using single limb calculations for sizes up to GCDEXT_ THRESHOLD, or double limb calculations above that. The single limb code is faster but doesn’t produce full-limb multipliers, hence not making full use of the mpn_addmul_1 calls.

When a CPU has a data-dependent multiplier, meaning one which is faster on operands with fewer bits, the extra work in the double-limb calculation might only save some looping overheads,

leading to a large GCDEXT_THRESHOLD.

Currently the single limb calculation doesn’t optimize for the small quotients that often occur, and this can lead to unusually low values of GCDEXT_THRESHOLD, depending on the CPU.

An analysis of double-limb calculations can be found in “A Double-Digit Lehmer-Euclid Algorithm” by Jebelean (see Appendix B [References], page 109). The code in GMP was developed independently.

It should be noted that when a double limb calculation is used, it’s used for the whole of that GCD, it doesn’t fall back to single limb part way through. This is because as the algorithm proceeds, the inputs a and b are reduced, but the cofactors x and y grow, so the multipliers at each step are applied to a roughly constant total number of limbs.

16.3.4 Jacobi Symbol

mpz_jacobi and mpz_kronecker are currently implemented with a simple binary algorithm similar to that described for the GCDs (see Section 16.3.1 [Binary GCD], page 91). They’re not very fast when both inputs are large. Lehmer’s multi-step improvement or a binary based multi-step algorithm is likely to be better.

When one operand fits a single limb, and that includes mpz_kronecker_ui and friends, an initial reduction is done with either mpn_mod_1 or mpn_modexact_1_odd, followed by the binary algorithm on a single limb. The binary algorithm is well suited to a single limb, and the whole calculation in this case is quite efficient.

In all the routines sign changes for the result are accumulated using some bit twiddling, avoiding table lookups or conditional jumps.

16.4 Powering Algorithms

16.4.1 Normal Powering

Normal mpz or mpf powering uses a simple binary algorithm, successively squaring and then multiplying by the base when a 1 bit is seen in the exponent, as per Knuth section 4.6.3. The “left to right” variant described there is used rather than algorithm A, since it’s just as easy and can be done with somewhat less temporary memory.

16.4.2 Modular Powering

Modular powering is implemented using a 2^{k}-ary sliding window algorithm, as per “Handbook of Applied Cryptography” algorithm 14.85 (see Appendix B [References], page 109). k is chosen according to the size of the exponent. Larger exponents use larger values of k, the choice being made to minimize the average number of multiplications that must supplement the squaring.

The modular multiplies and squares use either a simple division or the REDC method by Montgomery (see Appendix B [References], page 109). REDC is a little faster, essentially saving N single limb divisions in a fashion similar to an exact remainder (see Section 16.2.5 [Exact Remainder], page 90). The current REDC has some limitations. It’s only O(N^{2}) so above POWM_THRESHOLD division becomes faster and is used. It doesn’t attempt to detect small bases, but rather always uses a REDC form, which is usually a full size operand. And lastly it’s only applied to odd moduli.

16.5 Root Extraction Algorithms

16.5.1 Square Root

Square roots are taken using the “Karatsuba Square Root” algorithm by Paul Zimmermann (see Appendix B [References], page 109). This is expressed in a divide and conquer form, but as noted in the paper it can also be viewed as a discrete variant of Newton’s method.

In the Karatsuba multiplication range this is an O(\frac{3}{2}M(N/2)) algorithm, where M(n) is the time to multiply two numbers of n limbs. In the FFT multiplication range this grows to a bound of O(6M(N/2)). In practice a factor of about 1.5 to 1.8 is found in the Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range.

The algorithm does all its calculations in integers and the resulting mpn_sqrtrem is used for both mpz_sqrt and mpf_sqrt. The extended precision given by mpf_sqrt_ui is obtained by padding with zero limbs.

16.5.2 Nth Root

Integer Nth roots are taken using Newton’s method with the following iteration, where A is the input and n is the root to be taken.

a_{i+1} = \frac{1}{n} (\frac{A}{a_{i}^{n−1}} + (n − 1)a_{i})

The initial approximation a_{1} is generated bitwise by successively powering a trial root with or without new 1 bits, aiming to be just above the true root. The iteration converges quadratically when started from a good approximation. When n is large more initial bits are needed to get good convergence. The current implementation is not particularly well optimized.

16.5.3 Perfect Square

mpz_perfect_square_p is able to quickly exclude most non-squares by checking whether the input is a quadratic residue modulo some small integers.

The first test is modulo 256 which means simply examining the least significant byte. Only 44 different values occur as the low byte of a square, so 82.8% of non-squares can be immediately excluded. Similar tests modulo primes from 3 to 29 exclude 99.5% of those remaining, or if a limb is 64 bits then primes up to 53 are used, excluding 99.99%. A single N×1 remainder using PP from ‘gmp-impl.h’ quickly gives all these remainders.

A square root must still be taken for any value that passes the residue tests, to verify it’s really a square and not one of the 0.086% (or 0.000156% for 64 bits) non-squares that get through. See Section 16.5.1 [Square Root Algorithm], page 93.

16.5.4 Perfect Power

Detecting perfect powers is required by some factorization algorithms. Currently mpz_perfect_ power_p is implemented using repeated Nth root extractions, though naturally only prime roots need to be considered. (See Section 16.5.2 [Nth Root Algorithm], page 94.)

If a prime divisor p with multiplicity e can be found, then only roots which are divisors of e need to be considered, much reducing the work necessary. To this end divisibility by a set of small primes is checked.

16.6 Radix Conversion

Radix conversions are less important than other algorithms. A program dominated by conversions should probably use a different data representation.

16.6.1 Binary to Radix

Conversions from binary to a power-of-2 radix use a simple and fast O(N) bit extraction algorithm.

Conversions from binary to other radices use one of two algorithms. Sizes below GET_STR_ PRECOMPUTE_THRESHOLD use a basic O(N^{2}) method. Repeated divisions by b^{n} are made, where b is the radix and n is the biggest power that fits in a limb. But instead of simply using the remainder r from such divisions, an extra divide step is done to give a fractional limb representing r/b^{n}. The digits of r can then be extracted using multiplications by b rather than divisions. Special case code is provided for decimal, allowing multiplications by 10 to optimize to shifts and adds.

Above GET_STR_PRECOMPUTE_THRESHOLD a sub-quadratic algorithm is used. For an input t, powers b^{n2^{i}} of the radix are calculated, until a power between t and \sqrt{t} is reached. t is then divided by that largest power, giving a quotient which is the digits above that power, and a remainder which is those below. These two parts are in turn divided by the second highest power, and so on recursively. When a piece has been divided down to less than GET_STR_DC_THRESHOLD limbs, the basecase algorithm described above is used.

The advantage of this algorithm is that big divisions can make use of the sub-quadratic divide and conquer division (see Section 16.2.3 [Divide and Conquer Division], page 89), and big divisions tend to have less overheads than lots of separate single limb divisions anyway. But in any case the cost of calculating the powers b^{n2^{i}} must first be overcome.

GET_STR_PRECOMPUTE_THRESHOLD and GET_STR_DC_THRESHOLD represent the same basic thing,

the point where it becomes worth doing a big division to cut the input in half. GET_STR_ PRECOMPUTE_THRESHOLD includes the cost of calculating the radix power required, whereas GET_ STR_DC_THRESHOLD assumes that’s already available, which is the case when recursing.

Since the base case produces digits from least to most significant but they want to be stored from most to least, it’s necessary to calculate in advance how many digits there will be, or at least be sure not to underestimate that. For GMP the number of input bits is multiplied by chars_per_bit_exactly from mp_bases, rounding up. The result is either correct or one too big.

Examining some of the high bits of the input could increase the chance of getting the exact number of digits, but an exact result every time would not be practical, since in general the difference between numbers 100… and 99… is only in the last few bits and the work to identify 99… might well be almost as much as a full conversion.

mpf_get_str doesn’t currently use the algorithm described here, it multiplies or divides by a power of b to move the radix point to the just above the highest non-zero digit (or at worst one above that location), then multiplies by b^{n} to bring out digits. This is O(N^{2}) and is certainly not optimal.

The r/b^{n} scheme described above for using multiplications to bring out digits might be useful for more than a single limb. Some brief experiments with it on the base case when recursing didn’t give a noticable improvement, but perhaps that was only due to the implementation. Something similar would work for the sub-quadratic divisions too, though there would be the cost of calculating a bigger radix power.

Another possible improvement for the sub-quadratic part would be to arrange for radix powers that balanced the sizes of quotient and remainder produced, ie. the highest power would be an b^{nk} approximately equal to \sqrt{t}, not restricted to a 2^{i} factor. That ought to smooth out a graph of times against sizes, but may or may not be a net speedup.

16.6.2 Radix to Binary

Conversions from a power-of-2 radix into binary use a simple and fast O(N) bitwise concatenation algorithm.

Conversions from other radices use one of two algorithms. Sizes below SET_STR_THRESHOLD use a basic O(N^{2}) method. Groups of n digits are converted to limbs, where n is the biggest power of the base b which will fit in a limb, then those groups are accumulated into the result by multiplying by b^{n} and adding. This saves multi-precision operations, as per Knuth section 4.4 part E (see Appendix B [References], page 109). Some special case code is provided for decimal, giving the compiler a chance to optimize multiplications by 10.

Above SET_STR_THRESHOLD a sub-quadratic algorithm is used. First groups of n digits are converted into limbs. Then adjacent limbs are combined into limb pairs with xb^{n} + y, where x and y are the limbs. Adjacent limb pairs are combined into quads similarly with xb^{2n} + y. This continues until a single block remains, that being the result.

The advantage of this method is that the multiplications for each x are big blocks, allowing Karatsuba and higher algorithms to be used. But the cost of calculating the powers b^{n2^{i}} must be overcome. SET_STR_THRESHOLD usually ends up quite big, around 5000 digits, and on some processors much bigger still.

SET_STR_THRESHOLD is based on the input digits (and tuned for decimal), though it might be better based on a limb count, so as to be independent of the base. But that sort of count isn’t used by the base case and so would need some sort of initial calculation or estimate.

The main reason SET_STR_THRESHOLD is so much bigger than the corresponding GET_STR_

PRECOMPUTE_THRESHOLD is that mpn_mul_1 is much faster than mpn_divrem_1 (often by a factor

of 10, or more).

16.7 Other Algorithms

16.7.1 Factorial

Factorials n! are calculated by a simple product from 1 to n, but arranged into certain subproducts.

First as many factors as fit in a limb are accumulated, then two of those multiplied to give a 2-limb product. When two 2-limb products are ready they’re multiplied to a 4-limb product, and when two 4-limbs are ready they’re multiplied to an 8-limb product, etc. A stack of outstanding products is built up, with two of the same size multiplied together when ready.

Arranging for multiplications to have operands the same (or nearly the same) size means the Karatsuba and higher multiplication algorithms can be used. And even on sizes below the Karatsuba threshold an N×N multiply will give a basecase multiply more to work on.

An obvious improvement not currently implemented would be to strip factors of 2 from the products and apply them at the end with a bit shift. Another possibility would be to determine the prime factorization of the result (which can be done easily), and use a powering method, at each stage squaring then multiplying in those primes with a 1 in their exponent at that point. The advantage would be some multiplies turned into squares.

16.7.2 Binomial Coefficients

Binomial coefficients \binom{n}{k} are calculated by first arranging k ≤ n/2 using \binom{n}{k} = \binom{n}{n−k} if necessary, and then evaluating the following product simply from i = 2 to i = k.

\binom{n}{k} = (n − k + 1) ∏_{i=2}^{k} \frac{n − k + i}{i}

It’s easy to show that each denominator i will divide the product so far, so the exact division algorithm is used (see Section 16.2.4 [Exact Division], page 89).

The numerators n − k + i and denominators i are first accumulated into as many fit a limb, to save multi-precision operations, though for mpz_bin_ui this applies only to the divisors, since n is an mpz_t and n − k + i in general won’t fit in a limb at all.

An obvious improvement would be to strip factors of 2 from each multiplier and divisor and count them separately, to be applied with a bit shift at the end. Factors of 3 and perhaps 5 could even be handled similarly. Another possibility, if n is not too big, would be to determine the prime factorization of the result based on the factorials involved, and power up those primes appropriately. This would help most when k is near n/2.

16.7.3 Fibonacci Numbers

The Fibonacci functions mpz_fib_ui and mpz_fib2_ui are designed for calculating isolated F_{n} or F_{n},F_{n−1} values efficiently.

For small n, a table of single limb values in __gmp_fib_table is used. On a 32-bit limb this goes up to F_{47}, or on a 64-bit limb up to F_{93}. For convenience the table starts at F_{−1}.

Beyond the table, values are generated with a binary powering algorithm, calculating a pair F_{n} and F_{n−1} working from high to low across the bits of n. The formulas used are

\begin{aligned}F_{2k+1} & = 4F_{k}^{2} − F_{k−1}^{2} + 2(−1)^{k} \\ F_{2k−1} & = F_{k}^{2} + F_{k−1}^{2} \\ F_{2k} & = F_{2k+1} − F_{2k−1} \end{aligned}

At each step, k is the high b bits of n. If the next bit of n is 0 then F_{2k},F_{2k−1} is used, or if it’s a 1 then F_{2k+1},F_{2k} is used, and the process repeated until all bits of n are incorporated. Notice these formulas require just two squares per bit of n.

It’d be possible to handle the first few n above the single limb table with simple additions, using the defining Fibonacci recurrence F_{k+1} = F_{k} + F_{k−1}, but this is not done since it usually turns out to be faster for only about 10 or 20 values of n, and including a block of code for just those doesn’t seem worthwhile. If they really mattered it’d be better to extend the data table.

Using a table avoids lots of calculations on small numbers, and makes small n go fast. A bigger table would make more small n go fast, it’s just a question of balancing size against desired speed. For GMP the code is kept compact, with the emphasis primarily on a good powering algorithm.

mpz_fib2_ui returns both F_{n} and F_{n−1}, but mpz_fib_ui is only interested in F_{n}. In this case the last step of the algorithm can become one multiply instead of two squares. One of the following two formulas is used, according as n is odd or even.

\begin{aligned}F_{2k} & = F_{k}(F_{k} + 2F_{k−1}) \\ F_{2k+1} & = (2F_{k} + F_{k−1})(2F_{k} − F_{k−1}) + 2(−1)^{k} \end{aligned}

F_{2k+1} here is the same as above, just rearranged to be a multiply. For interest, the 2(−1)^{k} term both here and above can be applied just to the low limb of the calculation, without a carry or borrow into further limbs, which saves some code size. See comments with mpz_fib_ui and the internal mpn_fib2_ui for how this is done.

16.7.4 Lucas Numbers

mpz_lucnum2_ui derives a pair of Lucas numbers from a pair of Fibonacci numbers with the following simple formulas.

\begin{aligned}L_{k} & = F_{k} + 2F_{k−1} \\ L_{k−1} & = 2F_{k} − F_{k−1} \end{aligned}

mpz_lucnum_ui is only interested in L_{n}, and some work can be saved. Trailing zero bits on n can be handled with a single square each.

L_{2k} = L_{k}^{2} − 2(−1)^{k}

And the lowest 1 bit can be handled with one multiply of a pair of Fibonacci numbers, similar to what mpz_fib_ui does.

L_{2k+1} = 5F_{k−1}(2F_{k} + F_{k−1}) − 4(−1)^{k}

16.8 Assembler Coding

The assembler subroutines in GMP are the most significant source of speed at small to moderate sizes. At larger sizes algorithm selection becomes more important, but of course speedups in low level routines will still speed up everything proportionally.

Carry handling and widening multiplies that are important for GMP can’t be easily expressed in C. GCC asm blocks help a lot and are provided in ‘longlong.h’, but hand coding low level routines invariably offers a speedup over generic C by a factor of anything from 2 to 10.

16.8.1 Code Organisation

The various ‘mpn’ subdirectories contain machine-dependent code, written in C or assembler. The ‘mpn/generic’ subdirectory contains default code, used when there’s no machine-specific version of a particular file.

Each ‘mpn’ subdirectory is for an ISA family. Generally 32-bit and 64-bit variants in a family cannot share code and will have separate directories. Within a family further subdirectories may exist for CPU variants.

16.8.2 Assembler Basics

mpn_addmul_1 and mpn_submul_1 are the most important routines for overall GMP performance. All multiplications and divisions come down to repeated calls to these. mpn_add_n, mpn_sub_n, mpn_lshift and mpn_rshift are next most important.

On some CPUs assembler versions of the internal functions mpn_mul_basecase and mpn_sqr_ basecase give significant speedups, mainly through avoiding function call overheads. They can also potentially make better use of a wide superscalar processor.

The restrictions on overlaps between sources and destinations (see Chapter 8 [Low-level Functions], page 50) are designed to facilitate a variety of implementations. For example, knowing mpn_add_n won’t have partly overlapping sources and destination means reading can be done far ahead of writing on superscalar processors, and loops can be vectorized on a vector processor, depending on the carry handling.

16.8.3 Carry Propagation

The problem that presents most challenges in GMP is propagating carries from one limb to the next. In functions like mpn_addmul_1 and mpn_add_n, carries are the only dependencies between limb operations.

On processors with carry flags, a straightforward CISC style adc is generally best. AMD K6 mpn_addmul_1 however is an example of an unusual set of circumstances where a branch works out better.

On RISC processors generally an add and compare for overflow is used. This sort of thing can be seen in ‘mpn/generic/aors_n.c’. Some carry propagation schemes require 4 instructions, meaning at least 4 cycles per limb, but other schemes may use just 1 or 2. On wide superscalar processors performance may be completely determined by the number of dependent instructions between carry-in and carry-out for each limb.

On vector processors good use can be made of the fact that a carry bit only very rarely propagates more than one limb. When adding a single bit to a limb, there’s only a carry out if that limb was

0xFF...FF which on random data will be only 1 in 2mp bits per limb. ‘mpn/cray/add_n.c’ is

an example of this, it adds all limbs in parallel, adds one set of carry bits in parallel and then only rarely needs to fall through to a loop propagating further carries.

On the x86s, GCC (as of version 2.95.2) doesn’t generate particularly good code for the RISC style idioms that are necessary to handle carry bits in C. Often conditional jumps are generated where adc or sbb forms would be better. And so unfortunately almost any loop involving carry bits needs to be coded in assembler for best results.

16.8.4 Cache Handling

GMP aims to perform well both on operands that fit entirely in L1 cache and those which don’t.

Basic routines like mpn_add_n or mpn_lshift are often used on large operands, so L2 and main memory performance is important for them. mpn_mul_1 and mpn_addmul_1 are mostly used for multiply and square basecases, so L1 performance matters most for them, unless assembler versions of mpn_mul_basecase and mpn_sqr_basecase exist, in which case the remaining uses are mostly for larger operands.

For L2 or main memory operands, memory access times will almost certainly be more than the calculation time. The aim therefore is to maximize memory throughput, by starting a load of the next cache line which processing the contents of the previous one. Clearly this is only possible if the chip has a lock-up free cache or some sort of prefetch instruction. Most current chips have both these features.

Prefetching sources combines well with loop unrolling, since a prefetch can be initiated once per unrolled loop (or more than once if the loop covers more than one cache line).

On CPUs without write-allocate caches, prefetching destinations will ensure individual stores don’t go further down the cache hierarchy, limiting bandwidth. Of course for calculations which are slow anyway, like mpn_divrem_1, write-throughs might be fine.

The distance ahead to prefetch will be determined by memory latency versus throughput. The aim of course is to have data arriving continuously, at peak throughput. Some CPUs have limits on the number of fetches or prefetches in progress.

If a special prefetch instruction doesn’t exist then a plain load can be used, but in that case care must be taken not to attempt to read past the end of an operand, since that might produce a segmentation violation.

Some CPUs or systems have hardware that detects sequential memory accesses and initiates suitable cache movements automatically, making life easy.

16.8.5 Floating Point

Floating point arithmetic is used in GMP for multiplications on CPUs with poor integer multipliers. It’s mostly useful for mpn_mul_1, mpn_addmul_1 and mpn_submul_1 on 64-bit machines, and mpn_mul_basecase on both 32-bit and 64-bit machines.

With IEEE 53-bit double precision floats, integer multiplications producing up to 53 bits will give exact results. Breaking a 64×64 multiplication into eight 16×32 → 48 bit pieces is convenient. With some care though six 21×32 → 53 bit products can be used, if one of the lower two 21-bit pieces also uses the sign bit.

For the mpn_mul_1 family of functions on a 64-bit machine, the invariant single limb is split at the start, into 3 or 4 pieces. Inside the loop, the bignum operand is split into 32-bit pieces. Fast conversion of these unsigned 32-bit pieces to floating point is highly machine-dependent. In some cases, reading the data into the integer unit, zero-extending to 64-bits, then transferring to the floating point unit back via memory is the only option.

Converting partial products back to 64-bit limbs is usually best done as a signed conversion. Since all values are smaller than 2^{53}, signed and unsigned are the same, but most processors lack unsigned conversions.

Here is a diagram showing 16×32 bit products for an mpn_mul_1 or mpn_addmul_1 with a 64-bit limb. The single limb operand V is split into four 16-bit parts. The multi-limb operand U is split in the loop into two 32-bit parts.

v48 v32 v16 v00  V Operand
×  u32  u00  U Operand (one limb)
u00 × v00  p00 48-bit products
u00 × v16  p16
u00 × v32  p32
u00 × v48  p48
u32 × v00  r32
u32 × v16  r48
u32 × v32  r64
u32 × v48  r80

p32 and r32 can be summed using floating-point addition, and likewise p48 and r48. p00 and p16 can be summed with r64 and r80 from the previous iteration.

For each loop then, four 49-bit quantities are transfered to the integer unit, aligned as follows,

64 bits 64 bits

p00 + r64′  i00
p16 + r80′  i16
p32 + r32  i32
p48 + r48  i48

The challenge then is to sum these efficiently and add in a carry limb, generating a low 64-bit result limb and a high 33-bit carry limb (i48 extends 33 bits into the high half).

16.8.6 SIMD Instructions

The single-instruction multiple-data support in current microprocessors is aimed at signal processing algorithms where each data point can be treated more or less independently. There’s generally not much support for propagating the sort of carries that arise in GMP.

SIMD multiplications of say four 16×16 bit multiplies only do as much work as one 32×32 from GMP’s point of view, and need some shifts and adds besides. But of course if say the SIMD form is fully pipelined and uses less instruction decoding then it may still be worthwhile.

On the 80x86 chips, MMX has so far found a use in mpn_rshift and mpn_lshift since it allows 64-bit operations, and is used in a special case for 16-bit multipliers in the P55 mpn_mul_1. 3DNow and SSE haven’t found a use so far.

16.8.7 Software Pipelining

Software pipelining consists of scheduling instructions around the branch point in a loop. For example a loop taking a checksum of an array of limbs might have a load and an add, but the load wouldn’t be for that add, rather for the one next time around the loop. Each load then is effectively scheduled back in the previous iteration, allowing latency to be hidden.

Naturally this is wanted only when doing things like loads or multiplies that take a few cycles to complete, and only where a CPU has multiple functional units so that other work can be done while waiting.

A pipeline with several stages will have a data value in progress at each stage and each loop iteration moves them along one stage. This is like juggling.

Within the loop some moves between registers may be necessary to have the right values in the right places for each iteration. Loop unrolling can help this, with each unrolled block able to use different registers for different values, even if some shuffling is still needed just before going back to the top of the loop.

16.8.8 Loop Unrolling

Loop unrolling consists of replicating code so that several limbs are processed in each loop. At a minimum this reduces loop overheads by a corresponding factor, but it can also allow better register usage, for example alternately using one register combination and then another. Judicious use of m4 macros can help avoid lots of duplication in the source code.

Unrolling is commonly done to a power of 2 multiple so the number of unrolled loops and the number of remaining limbs can be calculated with a shift and mask. But other multiples can be used too, just by subtracting each n limbs processed from a counter and waiting for less than n remaining (or offsetting the counter by n so it goes negative when there’s less than n remaining).

The limbs not a multiple of the unrolling can be handled in various ways, for example

• A simple loop at the end (or the start) to process the excess. Care will be wanted that it isn’t too much slower than the unrolled part.

• A set of binary tests, for example after an 8-limb unrolling, test for 4 more limbs to process, then a further 2 more or not, and finally 1 more or not. This will probably take more code space than a simple loop.

• A switch statement, providing separate code for each possible excess, for example an 8-limb unrolling would have separate code for 0 remaining, 1 remaining, etc, up to 7 remaining. This might take a lot of code, but may be the best way to optimize all cases in combination with a deep pipelined loop.

• A computed jump into the middle of the loop, thus making the first iteration handle the excess. This should make times smoothly increase with size, which is attractive, but setups for the jump and adjustments for pointers can be tricky and could become quite difficult in combination with deep pipelining.

One way to write the setups and finishups for a pipelined unrolled loop is simply to duplicate the loop at the start and the end, then delete instructions at the start which have no valid antecedents, and delete instructions at the end whose results are unwanted. Sizes not a multiple of the unrolling can then be handled as desired.

17 Internals

This chapter is provided only for informational purposes and the various internals described here may change in future GMP releases. Applications expecting to be compatible with future releases should use only the documented interfaces described in previous chapters.

17.1 Integer Internals

mpz_t variables represent integers using sign and magnitude, in space dynamically allocated and reallocated. The fields are as follows.

_mp_size The number of limbs, or the negative of that when representing a negative integer. Zero is represented by _mp_size set to zero, in which case the _mp_d data is unused.

_mp_d A pointer to an array of limbs which is the magnitude. These are stored “little endian” as per the mpn functions, so _mp_d[0] is the least significant limb and _mp_ d[ABS(_mp_size)-1] is the most significant. Whenever _mp_size is non-zero, the most significant limb is non-zero. Currently there’s always at least one limb allocated, so for instance mpz_set_ui never needs to reallocate, and mpz_get_ui can fetch _mp_d[0] unconditionally (though its value is then only wanted if _mp_size is non-zero).

_mp_alloc

_mp_alloc is the number of limbs currently allocated at _mp_d, and naturally _mp_ alloc >= ABS(_mp_size). When an mpz routine is about to (or might be about to) increase _mp_size, it checks _mp_alloc to see whether there’s enough space, and reallocates if not. MPZ_REALLOC is generally used for this.

The various bitwise logical functions like mpz_and behave as if negative values were twos complement. But sign and magnitude is always used internally, and necessary adjustments are made during the calculations. Sometimes this isn’t pretty, but sign and magnitude are best for other routines.

Some internal temporary variables are setup with MPZ_TMP_INIT and these have _mp_d space obtained from TMP_ALLOC rather than the memory allocation functions. Care is taken to ensure that these are big enough that no reallocation is necessary (since it would have unpredictable consequences).

17.2 Rational Internals

mpq_t variables represent rationals using an mpz_t numerator and denominator (see Section 17.1 [Integer Internals], page 102).

The canonical form adopted is denominator positive (and non-zero), no common factors between numerator and denominator, and zero uniquely represented as 0/1.

It’s believed that casting out common factors at each stage of a calculation is best in general. A GCD is an O(N^{2}) operation so it’s better to do a few small ones immediately than to delay and have to do a big one later. Knowing the numerator and denominator have no common factors can be used for example in mpq_mul to make only two cross GCDs necessary, not four.

This general approach to common factors is badly sub-optimal in the presence of simple factorizations or little prospect for cancellation, but GMP has no way to know when this will occur. As per Section 3.10 [Efficiency], page 18, that’s left to applications. The mpq_t framework might still suit, with mpq_numref and mpq_denref for direct access to the numerator and denominator, or of course mpz_t variables can be used directly.

17.3 Float Internals

Efficient calculation is the primary aim of GMP floats and the use of whole limbs and simple rounding facilitates this.

mpf_t floats have a variable precision mantissa and a single machine word signed exponent. The mantissa is represented using sign and magnitude.

most significant limb least significant limb

       mp exp  →  mp d

· ← radix point

←  mp size  →

The fields are as follows.

_mp_size The number of limbs currently in use, or the negative of that when representing a negative value. Zero is represented by _mp_size and _mp_exp both set to zero, and in that case the _mp_d data is unused. (In the future _mp_exp might be undefined when representing zero.)

_mp_prec The precision of the mantissa, in limbs. In any calculation the aim is to produce _mp_prec limbs of result (the most significant being non-zero).

_mp_d A pointer to the array of limbs which is the absolute value of the mantissa. These are stored “little endian” as per the mpn functions, so _mp_d[0] is the least significant limb and _mp_d[ABS(_mp_size)-1] the most significant. The most significant limb is always non-zero, but there are no other restrictions on its value, in particular the highest 1 bit can be anywhere within the limb. _mp_prec+1 limbs are allocated to _mp_d, the extra limb being for convenience (see below). There are no reallocations during a calculation, only in a change of precision

with mpf_set_prec.

_mp_exp The exponent, in limbs, determining the location of the implied radix point. Zero means the radix point is just above the most significant limb. Positive values mean a radix point offset towards the lower limbs and hence a value ≥ 1, as for example in the diagram above. Negative exponents mean a radix point further above the highest limb. Naturally the exponent can be any value, it doesn’t have to fall within the limbs as the diagram shows, it can be a long way above or a long way below. Limbs other than those included in the {_mp_d,_mp_size} data are treated as zero.

The following various points should be noted.

Low Zeros The least significant limbs _mp_d[0] etc can be zero, though such low zeros can always be ignored. Routines likely to produce low zeros check and avoid them to save time in subsequent calculations, but for most routines they’re quite unlikely and aren’t checked.

Mantissa Size Range The _mp_size count of limbs in use can be less than _mp_prec if the value can be represented in less. This means low precision values or small integers stored in a high precision mpf_t can still be operated on efficiently. _mp_size can also be greater than _mp_prec. Firstly a value is allowed to use all of the _mp_prec+1 limbs available at _mp_d, and secondly when mpf_set_prec_raw lowers _mp_prec it leaves _mp_size unchanged and so the size can be arbitrarily bigger than _mp_prec.

Rounding All rounding is done on limb boundaries. Calculating _mp_prec limbs with the high non-zero will ensure the application requested minimum precision is obtained. The use of simple “trunc” rounding towards zero is efficient, since there’s no need to examine extra limbs and increment or decrement.

Bit Shifts Since the exponent is in limbs, there are no bit shifts in basic operations like mpf_ add and mpf_mul. When differing exponents are encountered all that’s needed is to adjust pointers to line up the relevant limbs. Of course mpf_mul_2exp and mpf_div_2exp will require bit shifts, but the choice is between an exponent in limbs which requires shifts there, or one in bits which requires them almost everywhere else.

Use of _mp_prec+1 Limbs

The extra limb on _mp_d (_mp_prec+1 rather than just _mp_prec) helps when an mpf routine might get a carry from its operation. mpf_add for instance will do an mpn_add of _mp_prec limbs. If there’s no carry then that’s the result, but if there is a carry then it’s stored in the extra limb of space and _mp_size becomes _mp_prec+1. Whenever _mp_prec+1 limbs are held in a variable, the low limb is not needed for the intended precision, only the _mp_prec high limbs. But zeroing it out or moving the rest down is unnecessary. Subsequent routines reading the value will simply take the high limbs they need, and this will be _mp_prec if their target has that same precision. This is no more than a pointer adjustment, and must be checked anyway since the destination precision can be different from the sources. Copy functions like mpf_set will retain a full _mp_prec+1 limbs if available. This ensures that a variable which has _mp_size equal to _mp_prec+1 will get its full exact value copied. Strictly speaking this is unnecessary since only _mp_prec limbs are needed for the application’s requested precision, but it’s considered that an mpf_ set from one variable into another of the same precision ought to produce an exact copy.

Application Precisions __GMPF_BITS_TO_PREC converts an application requested precision to an _mp_prec. The value in bits is rounded up to a whole limb then an extra limb is added since the most significant limb of _mp_d is only non-zero and therefore might contain only one bit. __GMPF_PREC_TO_BITS does the reverse conversion, and removes the extra limb from _mp_prec before converting to bits. The net effect of reading back with mpf_get_ prec is simply the precision rounded up to a multiple of mp_bits_per_limb. Note that the extra limb added here for the high only being non-zero is in addition to the extra limb allocated to _mp_d. For example with a 32-bit limb, an application request for 250 bits will be rounded up to 8 limbs, then an extra added for the high being only non-zero, giving an _mp_prec of 9. _mp_d then gets 10 limbs allocated. Reading back with mpf_get_prec will take _mp_prec subtract 1 limb and multiply by 32, giving 256 bits. Strictly speaking, the fact the high limb has at least one bit means that a float with, say, 3 limbs of 32-bits each will be holding at least 65 bits, but for the purposes of mpf_t it’s considered simply to be 64 bits, a nice multiple of the limb size.

17.4 Raw Output Internals

mpz_out_raw uses the following format.

size  data bytes

The size is 4 bytes written most significant byte first, being the number of subsequent data bytes, or the twos complement negative of that when a negative integer is represented. The data bytes are the absolute value of the integer, written most significant byte first.

The most significant data byte is always non-zero, so the output is the same on all systems, irrespective of limb size.

In GMP 1, leading zero bytes were written to pad the data bytes to a multiple of the limb size. mpz_inp_raw will still accept this, for compatibility.

The use of “big endian” for both the size and data fields is deliberate, it makes the data easy to read in a hex dump of a file. Unfortunately it also means that the limb data must be reversed when reading or writing, so neither a big endian nor little endian system can just read and write

_mp_d.

17.5 C++ Interface Internals

A system of expression templates is used to ensure something like a=b+c turns into a simple call to mpz_add etc. For mpf_class and mpfr_class the scheme also ensures the precision of the final destination is used for any temporaries within a statement like f=w*x+y*z. These are important features which a naive implementation cannot provide.

A simplified description of the scheme follows. The true scheme is complicated by the fact that expressions have different return types. For detailed information, refer to the source code.

To perform an operation, say, addition, we first define a “function object” evaluating it,

     struct __gmp_binary_plus
     {
       static void eval(mpf_t f, mpf_t g, mpf_t h) { mpf_add(f, g, h); }
     };

And an “additive expression” object,

     __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >
     operator+(const mpf_class &f, const mpf_class &g)
     {
       return __gmp_expr
         <__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >(f, g);
     }

The seemingly redundant __gmp_expr<__gmp_binary_expr<...>> is used to encapsulate any possible kind of expression into a single template type. In fact even mpf_class etc are typedef specializations of __gmp_expr.

Next we define assignment of __gmp_expr to mpf_class.

     template <class T>
     mpf_class & mpf_class::operator=(const __gmp_expr<T> &expr)
     {
       expr.eval(this->get_mpf_t(), this->precision());
       return *this;
     }
     template <class Op>
     void __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, Op> >::eval
     (mpf_t f, unsigned long int precision)
     {
       Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t());

}

where expr.val1 and expr.val2 are references to the expression’s operands (here expr is the

__gmp_binary_expr stored within the __gmp_expr).

This way, the expression is actually evaluated only at the time of assignment, when the required precision (that of f) is known. Furthermore the target mpf_t is now available, thus we can call mpf_add directly with f as the output argument.

Compound expressions are handled by defining operators taking subexpressions as their arguments, like this:

template <class T, class U>
__gmp_expr
<__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
operator+(const __gmp_expr<T> &expr1, const __gmp_expr<U> &expr2)
{
  return __gmp_expr
    <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
    (expr1, expr2);
}

And the corresponding specializations of __gmp_expr::eval:

template <class T, class U, class Op>
void __gmp_expr
<__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, Op> >::eval
(mpf_t f, unsigned long int precision)
{
  // declare two temporaries
  mpf_class temp1(expr.val1, precision), temp2(expr.val2, precision);
  Op::eval(f, temp1.get_mpf_t(), temp2.get_mpf_t());
}

The expression is thus recursively evaluated to any level of complexity and all subexpressions are evaluated to the precision of f.

Appendix A Contributors

Torbjorn Granlund wrote the original GMP library and is still developing and maintaining it. Several other individuals and organizations have contributed to GMP in various ways. Here is a list in chronological order:

Gunnar Sjoedin and Hans Riesel helped with mathematical problems in early versions of the library.

Richard Stallman contributed to the interface design and revised the first version of this manual.

Brian Beuning and Doug Lea helped with testing of early versions of the library and made creative suggestions.

John Amanatides of York University in Canada contributed the function mpz_probab_prime_p.

Paul Zimmermann of Inria sparked the development of GMP 2, with his comparisons between bignum packages.

Ken Weber (Kent State University, Universidade Federal do Rio Grande do Sul) contributed mpz_gcd, mpz_divexact, mpn_gcd, and mpn_bdivmod, partially supported by CNPq (Brazil) grant 301314194-2.

Per Bothner of Cygnus Support helped to set up GMP to use Cygnus’ configure. He has also made valuable suggestions and tested numerous intermediary releases.

Joachim Hollman was involved in the design of the mpf interface, and in the mpz design revisions for version 2.

Bennet Yee contributed the initial versions of mpz_jacobi and mpz_legendre.

Andreas Schwab contributed the files ‘mpn/m68k/lshift.S’ and ‘mpn/m68k/rshift.S’ (now in ‘.asm’ form).

The development of floating point functions of GNU MP 2, were supported in part by the ESPRIT-BRA (Basic Research Activities) 6846 project POSSO (POlynomial System SOlving).

GNU MP 2 was finished and released by SWOX AB, SWEDEN, in cooperation with the IDA Center for Computing Sciences, USA.

Robert Harley of Inria, France and David Seal of ARM, England, suggested clever improvements for population count.

Robert Harley also wrote highly optimized Karatsuba and 3-way Toom multiplication functions for GMP 3. He also contributed the ARM assembly code.

Torsten Ekedahl of the Mathematical department of Stockholm University provided significant inspiration during several phases of the GMP development. His mathematical expertise helped improve several algorithms.

Paul Zimmermann wrote the Divide and Conquer division code, the REDC code, the REDCbased mpz powm code, the FFT multiply code, and the Karatsuba square root. The ECMNET project Paul is organizing was a driving force behind many of the optimizations in GMP 3.

Linus Nordberg wrote the new configure system based on autoconf and implemented the new random functions.

Kent Boortz made the Macintosh port.

Kevin Ryde worked on a number of things: optimized x86 code, m4 asm macros, parameter tuning, speed measuring, the configure system, function inlining, divisibility tests, bit scanning, Jacobi symbols, Fibonacci and Lucas number functions, printf and scanf functions, perl interface, demo expression parser, the algorithms chapter in the manual, ‘gmpasm-mode.el’, and various miscellaneous improvements elsewhere. Steve Root helped write the optimized alpha 21264 assembly code. Gerardo Ballabio wrote the ‘gmpxx.h’ C++ class interface and the C++ istream input routines. GNU MP 4.0 was finished and released by Torbjorn Granlund and Kevin Ryde. Torbjorn’s work was partially funded by the IDA Center for Computing Sciences, USA. (This list is chronological, not ordered after significance. If you have contributed to GMP but are not listed above, please tell tege@swox.com about the omission!) Thanks goes to Hans Thorsen for donating an SGI system for the GMP test system environment.

Appendix B References

B.1 Books

• Jonathan M. Borwein and Peter B. Borwein, “Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity”, Wiley, John & Sons, 1998.

• Henri Cohen, “A Course in Computational Algebraic Number Theory”, Graduate Texts in Mathematics number 138, Springer-Verlag, 1993.

 http://www.math.u-bordeaux.fr/~cohen

• Donald E. Knuth, “The Art of Computer Programming”, volume 2, “Seminumerical Algorithms”, 3rd edition, Addison-Wesley, 1998.

 http://www-cs-faculty.stanford.edu/~knuth/taocp.html

• John D. Lipson, “Elements of Algebra and Algebraic Computing”, The Benjamin Cummings Publishing Company Inc, 1981.

• Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, “Handbook of Applied

Cryptography”, http://www.cacr.math.uwaterloo.ca/hac/

• Richard M. Stallman, “Using and Porting GCC”, Free Software Foundation, 1999, avail-

able online http://www.gnu.org/software/gcc/onlinedocs/, and in the GCC package
ftp://ftp.gnu.org/gnu/gcc/

B.2 Papers

• Christoph Burnikel and Joachim Ziegler, “Fast Recursive Division”, Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022,

 http://data.mpi-sb.mpg.de/internet/reports.nsf/NumberView/1998-1-022

• Torbjorn Granlund and Peter L. Montgomery, “Division by Invariant Integers using Multiplication”, in Proceedings of the SIGPLAN PLDI’94 Conference, June 1994. Also available

ftp://ftp.cwi.nl/pub/pmontgom/divcnst.psa4.gz (and .psl.gz).

• Peter L. Montgomery, “Modular Multiplication Without Trial Division”, in Mathematics of Computation, volume 44, number 170, April 1985.

• Tudor Jebelean, “An algorithm for exact division”, Journal of Symbolic Computation, volume 15, 1993, pp. 169-180. Research report version available

 ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz

• Tudor Jebelean, “Exact Division with Karatsuba Complexity - Extended Abstract”, RISC-Linz technical report 96-31,

 ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-31.ps.gz

• Tudor Jebelean, “Practical Integer Division with Karatsuba Complexity”, ISSAC 97, pp. 339-341. Technical report available

 ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-29.ps.gz

• Tudor Jebelean, “A Generalization of the Binary GCD Algorithm”, ISSAC 93, pp. 111-116. Technical report version available

 ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-01.ps.gz

• Tudor Jebelean, “A Double-Digit Lehmer-Euclid Algorithm for Finding the GCD of Long Integers”, Journal of Symbolic Computation, volume 19, 1995, pp. 145-157. Technical report version also available

 ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz

• Werner Krandick and Tudor Jebelean, “Bidirectional Exact Integer Division”, Journal of Symbolic Computation, volume 21, 1996, pp. 441-455. Early technical report version also

available ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1994/94-50.ps.gz

• R. Moenck and A. Borodin, “Fast Modular Transforms via Division”, Proceedings of the 13th Annual IEEE Symposium on Switching and Automata Theory, October 1972, pp. 90-96. Reprinted as “Fast Modular Transforms”, Journal of Computer and System Sciences, volume 8, number 3, June 1974, pp. 366-386.

• Arnold Schönhage and Volker Strassen, “Schnelle Multiplikation grosser Zahlen”, Computing 7, 1971, pp. 281-292.

• Kenneth Weber, “The accelerated integer GCD algorithm”, ACM Transactions on Mathematical Software, volume 21, number 1, March 1995, pp. 111-122.

• Paul Zimmermann, “Karatsuba Square Root”, INRIA Research Report 3805, November

1999, http://www.inria.fr/RRRT/RR-3805.html

• Paul Zimmermann, “A Proof of GMP Fast Division and Square Root Implementations”,

 http://www.loria.fr/~zimmerma/papers/proof-div-sqrt.ps.gz

• Dan Zuras, “On Squaring and Multiplying Large Integers”, ARITH-11: IEEE Symposium on Computer Arithmetic, 1993, pp. 260 to 271. Reprinted as “More on Multiplying and Squaring Large Integers”, IEEE Transactions on Computers, volume 43, number 8, August 1994, pp. 899-908.

Appendix C GNU Free Documentation License

Version 1.1, March 2000

Copyright \begin{matrix}c \\ ◯ \end{matrix} 2000 Free Software Foundation, Inc. 59 Temple Place, Suite 330, Boston, MA 02111-1307, USA

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Concept Index 117

Concept Index

A

ABI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 About this manual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 alloca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Allocation of memory . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Anonymous FTP of latest version . . . . . . . . . . . . . . . . 2 Application Binary Interface . . . . . . . . . . . . . . . . . . . . . 8 Arithmetic functions. . . . . . . . . . . . . . . . . . . . . 28, 40, 47 Assignment functions . . . . . . . . . . . . . . . . . . . . . . . 26, 45 Autoconf detections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

B

Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Berkeley MP compatible functions. . . . . . . . . . . . . . . 76 Binomial coefficient functions . . . . . . . . . . . . . . . . . . . 33 Bit manipulation functions. . . . . . . . . . . . . . . . . . . . . . 34 Bit shift left . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Bit shift right . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Bits per limb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 BSD MP compatible functions . . . . . . . . . . . . . . . . . . 76 Bug reporting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Build notes for binary packaging . . . . . . . . . . . . . . . . 10 Build notes for particular systems . . . . . . . . . . . . . . . 10 Build options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Build problems known . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Building GMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

C

C++ Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 C++ istream input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 C++ ostream output. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Comparison functions. . . . . . . . . . . . . . . . . . . . 34, 41, 48 Compatibility with older versions. . . . . . . . . . . . . . . . 18 Conditions for copying GNU MP . . . . . . . . . . . . . . . . . 1 Configuring GMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Contributors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Conventions for parameters . . . . . . . . . . . . . . . . . . . . . 16 Conventions for variables . . . . . . . . . . . . . . . . . . . . . . . 15 Conversion functions . . . . . . . . . . . . . . . . . . . . 27, 40, 46 Copying conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 CPUs supported . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Custom allocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

D

Debugging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Demonstration programs. . . . . . . . . . . . . . . . . . . . . . . . . 7 ‘DESTDIR’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Division algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Division functions . . . . . . . . . . . . . . . . . . . . . . . 29, 40, 47

E

Efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Exact division functions . . . . . . . . . . . . . . . . . . . . . . . . 30 Example programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Exponentiation functions . . . . . . . . . . . . . . . . . . . . 31, 47 Export . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Extended GCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

F

Factorial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 FDL, GNU Free Documentation License . . . . . . . . 111 Fibonacci sequence functions . . . . . . . . . . . . . . . . . . . . 33 Float arithmetic functions . . . . . . . . . . . . . . . . . . . . . . 47 Float assignment functions. . . . . . . . . . . . . . . . . . . . . . 45 Float comparison functions . . . . . . . . . . . . . . . . . . . . . 48 Float conversion functions . . . . . . . . . . . . . . . . . . . . . . 46 Float functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Float init and assign functions . . . . . . . . . . . . . . . . . . 45 Float initialization functions . . . . . . . . . . . . . . . . . . . . 43 Float input and output functions . . . . . . . . . . . . . . . . 48 Float miscellaneous functions . . . . . . . . . . . . . . . . . . . 49 Float sign tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Floating-point functions . . . . . . . . . . . . . . . . . . . . . . . . 43 Floating-point number. . . . . . . . . . . . . . . . . . . . . . . . . . 14 Formatted input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Formatted output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 FTP of latest version . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Function classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

G

GMP version number. . . . . . . . . . . . . . . . . . . . . . . . . . . 17 ‘gmp.h’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 gmpxx.h. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 GNU Free Documentation License . . . . . . . . . . . . . . 111 Greatest common divisor algorithms . . . . . . . . . . . . . 91 Greatest common divisor functions . . . . . . . . . . . . . . 32

H

Headers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Home page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

I

118 GNU MP 4.1

I/O functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 35, 42, 48 Import . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Initialization and assignment functions …. 27, 39, 45 Initialization functions . . . . . . . . . . . . . . . . . . . . . . 25, 43 Input functions. . . . . . . . . . . . . . . . . . . . . . . . . . 35, 42, 48 Installing GMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Instruction Set Architecture. . . . . . . . . . . . . . . . . . . . . . 8 Integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Integer arithmetic functions . . . . . . . . . . . . . . . . . . . . . 28 Integer assignment functions . . . . . . . . . . . . . . . . . . . . 26 Integer bit manipulation functions . . . . . . . . . . . . . . . 34 Integer comparison functions . . . . . . . . . . . . . . . . . . . . 34 Integer conversion functions. . . . . . . . . . . . . . . . . . . . . 27 Integer division functions . . . . . . . . . . . . . . . . . . . . . . . 29 Integer exponentiation functions. . . . . . . . . . . . . . . . . 31 Integer export . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Integer functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Integer import . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Integer init and assign . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Integer initialization functions. . . . . . . . . . . . . . . . . . . 25 Integer input and output functions . . . . . . . . . . . . . . 35 Integer miscellaneous functions . . . . . . . . . . . . . . . . . . 38 Integer random number functions. . . . . . . . . . . . . . . . 36 Integer root functions . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Integer sign tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 ISA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 istream input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

J

Jacobi symbol functions . . . . . . . . . . . . . . . . . . . . . . . . 32

K

Kronecker symbol functions . . . . . . . . . . . . . . . . . . . . . 33

L

Latest version of GMP . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Least common multiple functions . . . . . . . . . . . . . . . . 32 Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Libtool versioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 License conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Limb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Limb size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Linking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Logical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Low-level functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Lucas number functions . . . . . . . . . . . . . . . . . . . . . . . . 33

M

Mailing list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Memory allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Memory Management . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Miscellaneous float functions . . . . . . . . . . . . . . . . . . . . 49 Miscellaneous integer functions . . . . . . . . . . . . . . . . . . 38 Modular inverse functions. . . . . . . . . . . . . . . . . . . . . . . 32 ‘mp.h’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 MPFR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 mpfrxx.h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Multi-threading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Multiplication algorithms . . . . . . . . . . . . . . . . . . . . . . . 82

N

Nails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Nomenclature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Number theoretic functions . . . . . . . . . . . . . . . . . . . . . 32 Numerator and denominator . . . . . . . . . . . . . . . . . . . . 41

O

ostream output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Output functions . . . . . . . . . . . . . . . . . . . . . . . . 35, 42, 48

P

Packaged builds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Parameter conventions. . . . . . . . . . . . . . . . . . . . . . . . . . 16 Particular systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Powering algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Powering functions. . . . . . . . . . . . . . . . . . . . . . . . . . 31, 47 Precision of floats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Prime testing functions . . . . . . . . . . . . . . . . . . . . . . . . . 32 printf formatted output . . . . . . . . . . . . . . . . . . . . . . . 60 Profiling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

R

Radix conversion algorithms . . . . . . . . . . . . . . . . . . . . 94 Random number functions. . . . . . . . . . . . . . . . . . . 36, 58 Random number seeding. . . . . . . . . . . . . . . . . . . . . . . . 59 Random number state . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Rational arithmetic functions . . . . . . . . . . . . . . . . . . . 40 Rational comparison functions . . . . . . . . . . . . . . . . . . 41 Rational conversion functions . . . . . . . . . . . . . . . . . . . 40 Rational init and assign . . . . . . . . . . . . . . . . . . . . . . . . 39 Rational input and output functions . . . . . . . . . . . . . 42 Rational number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Rational number functions . . . . . . . . . . . . . . . . . . . . . . 39 Rational numerator and denominator . . . . . . . . . . . . 41 Rational sign tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Reentrancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Reporting bugs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Root extraction algorithms . . . . . . . . . . . . . . . . . . . . . 93 Root extraction functions . . . . . . . . . . . . . . . . . . . 31, 47

S

scanf formatted input . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Shared library versioning . . . . . . . . . . . . . . . . . . . . . . . 10 Sign tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34, 41, 48 Stack overflow segfaults. . . . . . . . . . . . . . . . . . . . . . . . . . 6 Stripped libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

T

Thread safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

U

Upward compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Useful macros and constants . . . . . . . . . . . . . . . . . . . . 17 User-defined precision . . . . . . . . . . . . . . . . . . . . . . . . . . 43

V

Variable conventions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Version number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

W

Web page. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Function and Type Index 119

Function and Type Index

*

*mpz_export. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

__GNU_MP_VERSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 __GNU_MP_VERSION_MINOR . . . . . . . . . . . . . . . . . . . . . . . 17

__GNU_MP_VERSION_PATCHLEVEL . . . . . . . . . . . . . . . . . 17

_mpz_realloc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

A

abs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70, 71, 72 allocate_function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

C

ceil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 cmp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70, 71, 72

D

deallocate_function . . . . . . . . . . . . . . . . . . . . . . . . . . 78

F

floor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

G

gcd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 gmp_asprintf. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 gmp_fprintf. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 gmp_fscanf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 GMP_LIMB_BITS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 GMP_NAIL_BITS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 GMP_NAIL_MASK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 GMP_NUMB_BITS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 GMP_NUMB_MASK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 GMP_NUMB_MAX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 gmp_obstack_printf . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 gmp_obstack_vprintf . . . . . . . . . . . . . . . . . . . . . . . . . . 63 gmp_printf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 gmp_randclass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 gmp_randclass::get_f . . . . . . . . . . . . . . . . . . . . . . . . . 73

gmp_randclass::get_z_bits. . . . . . . . . . . . . . . . . . . . 73
gmp_randclass::get_z_range . . . . . . . . . . . . . . . . . . 73
gmp_randclass::gmp_randclass . . . . . . . . . . . . . . . . 73

gmp_randclass::seed . . . . . . . . . . . . . . . . . . . . . . . . . . 73 gmp_randclear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 gmp_randinit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 gmp_randinit_default . . . . . . . . . . . . . . . . . . . . . . . . . 58 gmp_randinit_lc_2exp . . . . . . . . . . . . . . . . . . . . . . . . . 58

gmp_randinit_lc_2exp_size. . . . . . . . . . . . . . . . . . . . 58

gmp_randseed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 gmp_randseed_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 gmp_scanf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 gmp_snprintf. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 gmp_sprintf. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 gmp_sscanf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 gmp_vasprintf. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 gmp_version. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 gmp_vfprintf. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 gmp_vfscanf. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 gmp_vprintf. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 gmp_vscanf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 gmp_vsnprintf. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 gmp_vsprintf. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 gmp_vsscanf. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

H

hypot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

I

itom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

M

madd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 mcmp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 mdiv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 mfree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 min . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 mout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 mp_bits_per_limb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 mp_limb_t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 mp_set_memory_functions . . . . . . . . . . . . . . . . . . . . . . 78 mpf_abs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 mpf_add . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 mpf_add_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 mpf_ceil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 mpf_class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 mpf_class::fits_sint_p . . . . . . . . . . . . . . . . . . . . . . . 72 mpf_class::fits_slong_p . . . . . . . . . . . . . . . . . . . . . . 72

mpf_class::fits_sshort_p. . . . . . . . . . . . . . . . . . . . . 72

mpf_class::fits_uint_p . . . . . . . . . . . . . . . . . . . . . . . 72 mpf_class::fits_ulong_p . . . . . . . . . . . . . . . . . . . . . . 72

mpf_class::fits_ushort_p. . . . . . . . . . . . . . . . . . . . . 72

mpf_class::get_d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 mpf_class::get_mpf_t . . . . . . . . . . . . . . . . . . . . . . . . . 69 mpf_class::get_prec . . . . . . . . . . . . . . . . . . . . . . . . . . 72 mpf_class::get_si . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 mpf_class::get_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 mpf_class::mpf_class . . . . . . . . . . . . . . . . . . . . . . . . . 72 mpf_class::set_prec . . . . . . . . . . . . . . . . . . . . . . . . . . 72 mpf_class::set_prec_raw . . . . . . . . . . . . . . . . . . . . . . 72 mpf_clear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 mpf_cmp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 mpf_cmp_d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 mpf_cmp_si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 mpf_cmp_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 mpf_div . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 mpf_div_2exp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 mpf_div_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 mpf_eq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 mpf_fits_sint_p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 mpf_fits_slong_p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 mpf_fits_sshort_p . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 mpf_fits_uint_p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 mpf_fits_ulong_p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 mpf_fits_ushort_p . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 mpf_floor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 mpf_get_d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 mpf_get_d_2exp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 mpf_get_default_prec . . . . . . . . . . . . . . . . . . . . . . . . . 43 mpf_get_prec. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 mpf_get_si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 mpf_get_str. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 mpf_get_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 mpf_init . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 mpf_init_set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 mpf_init_set_d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 mpf_init_set_si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 mpf_init_set_str . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 mpf_init_set_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 mpf_init2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 mpf_inp_str. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 mpf_integer_p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 mpf_mul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 mpf_mul_2exp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 mpf_mul_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 mpf_neg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 mpf_out_str. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 mpf_pow_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 mpf_random2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 mpf_reldiff. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 mpf_set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 mpf_set_d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 mpf_set_default_prec . . . . . . . . . . . . . . . . . . . . . . . . . 43 mpf_set_prec. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 mpf_set_prec_raw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 mpf_set_q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 mpf_set_si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 mpf_set_str. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 mpf_set_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 mpf_set_z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 mpf_sgn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 mpf_sqrt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 mpf_sqrt_ui. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 mpf_sub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 mpf_sub_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 mpf_swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 mpf_t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 mpf_trunc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 mpf_ui_div . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 mpf_ui_sub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 mpf_urandomb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 mpfr_class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 mpn_add . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 mpn_add_1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 mpn_add_n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 mpn_addmul_1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 mpn_bdivmod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 mpn_cmp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 mpn_divexact_by3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 mpn_divexact_by3c . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 mpn_divmod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 mpn_divmod_1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 mpn_divrem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 mpn_divrem_1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 mpn_gcd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 mpn_gcd_1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 mpn_gcdext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 mpn_get_str. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 mpn_hamdist. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 mpn_lshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 mpn_mod_1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 mpn_mul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 mpn_mul_1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 mpn_mul_n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 mpn_perfect_square_p . . . . . . . . . . . . . . . . . . . . . . . . . 56 mpn_popcount. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 mpn_random . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 mpn_random2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 mpn_rshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 mpn_scan0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 mpn_scan1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 mpn_set_str. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 mpn_sqrtrem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 mpn_sub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 mpn_sub_1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 mpn_sub_n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 mpn_submul_1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 mpn_tdiv_qr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 mpq_abs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 mpq_add . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 mpq_canonicalize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 mpq_class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 mpq_class::canonicalize . . . . . . . . . . . . . . . . . . . . . . 71 mpq_class::get_d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 mpq_class::get_den . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

                                                  mpq_class::get_den_mpz_t. . . . . . . . . . . . . . . . . . . . . 71

mpq_class::get_mpq_t . . . . . . . . . . . . . . . . . . . . . . . . . 69 mpq_class::get_num . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

                                                  mpq_class::get_num_mpz_t. . . . . . . . . . . . . . . . . . . . . 71

mpq_class::mpq_class. . . . . . . . . . . . . . . . . . . . . . 70, 71 mpq_clear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 mpq_cmp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 mpq_cmp_si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 mpq_cmp_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 mpq_denref . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 mpq_div . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 mpq_div_2exp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 mpq_equal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 mpq_get_d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 mpq_get_den. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 mpq_get_num. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 mpq_get_str. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 mpq_init . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 mpq_inp_str. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 mpq_inv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 mpq_mul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 mpq_mul_2exp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 mpq_neg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 mpq_numref . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 mpq_out_str. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 mpq_set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 mpq_set_d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 mpq_set_den. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 mpq_set_f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 mpq_set_num. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 mpq_set_si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 mpq_set_str. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 mpq_set_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 mpq_set_z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 mpq_sgn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 mpq_sub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 mpq_swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 mpq_t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 mpz_abs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_add . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 mpz_add_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 mpz_addmul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 mpz_addmul_ui. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 mpz_and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 mpz_array_init . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Function and Type Index 121

mpz_bin_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 mpz_bin_uiui. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 mpz_cdiv_q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_cdiv_q_2exp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_cdiv_q_ui. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_cdiv_qr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_cdiv_qr_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_cdiv_r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_cdiv_r_2exp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_cdiv_r_ui. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_cdiv_ui. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 mpz_class::fits_sint_p . . . . . . . . . . . . . . . . . . . . . . . 70 mpz_class::fits_slong_p . . . . . . . . . . . . . . . . . . . . . . 70

mpz_class::fits_sshort_p. . . . . . . . . . . . . . . . . . . . . 70

mpz_class::fits_uint_p . . . . . . . . . . . . . . . . . . . . . . . 70 mpz_class::fits_ulong_p . . . . . . . . . . . . . . . . . . . . . . 70

mpz_class::fits_ushort_p. . . . . . . . . . . . . . . . . . . . . 70

mpz_class::get_d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 mpz_class::get_mpz_t . . . . . . . . . . . . . . . . . . . . . . . . . 69 mpz_class::get_si . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 mpz_class::get_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 mpz_class::mpz_class . . . . . . . . . . . . . . . . . . . . . . . . . 69 mpz_clear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 mpz_clrbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 mpz_cmp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 mpz_cmp_d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 mpz_cmp_si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 mpz_cmp_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 mpz_cmpabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 mpz_cmpabs_d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 mpz_cmpabs_ui. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 mpz_com . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 mpz_congruent_2exp_p . . . . . . . . . . . . . . . . . . . . . . . . . 31 mpz_congruent_p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 mpz_congruent_ui_p . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 mpz_divexact. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 mpz_divexact_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 mpz_divisible_2exp_p . . . . . . . . . . . . . . . . . . . . . . . . . 30 mpz_divisible_p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 mpz_divisible_ui_p . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 mpz_even_p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 mpz_fac_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 mpz_fdiv_q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_fdiv_q_2exp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_fdiv_q_ui. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_fdiv_qr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_fdiv_qr_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_fdiv_r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_fdiv_r_2exp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_fdiv_r_ui. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_fdiv_ui. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_fib_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 mpz_fib2_ui. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 mpz_fits_sint_p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 mpz_fits_slong_p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 mpz_fits_sshort_p . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 mpz_fits_uint_p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 mpz_fits_ulong_p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 mpz_fits_ushort_p . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 mpz_gcd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 mpz_gcd_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 mpz_gcdext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 mpz_get_d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 mpz_get_d_2exp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 mpz_get_si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 mpz_get_str. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 mpz_get_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 mpz_getlimbn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 mpz_hamdist. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 mpz_import . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 mpz_init . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 mpz_init_set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 mpz_init_set_d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 mpz_init_set_si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 mpz_init_set_str . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 mpz_init_set_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 mpz_init2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 mpz_inp_raw. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 mpz_inp_str. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 mpz_invert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 mpz_ior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 mpz_jacobi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 mpz_kronecker. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 mpz_kronecker_si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 mpz_kronecker_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 mpz_lcm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 mpz_lcm_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 mpz_legendre. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 mpz_lucnum_ui. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 mpz_lucnum2_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 mpz_mod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 mpz_mod_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 mpz_mul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 mpz_mul_2exp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_mul_si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 mpz_mul_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 mpz_neg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_nextprime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 mpz_odd_p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 mpz_out_raw. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 mpz_out_str. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 mpz_perfect_power_p . . . . . . . . . . . . . . . . . . . . . . . . . . 31 mpz_perfect_square_p . . . . . . . . . . . . . . . . . . . . . . . . . 31 mpz_popcount. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 mpz_pow_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 mpz_powm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 mpz_powm_ui. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 mpz_probab_prime_p . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 mpz_random . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 mpz_random2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 mpz_realloc2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 mpz_remove . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 mpz_root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 mpz_rrandomb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 mpz_scan0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 mpz_scan1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 mpz_set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 mpz_set_d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 mpz_set_f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 mpz_set_q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 mpz_set_si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 mpz_set_str. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 mpz_set_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 mpz_setbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 mpz_sgn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 mpz_si_kronecker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 mpz_size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 mpz_sizeinbase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 mpz_sqrt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 mpz_sqrtrem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 mpz_sub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 mpz_sub_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 mpz_submul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 mpz_submul_ui. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 mpz_swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 mpz_t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 mpz_tdiv_q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_tdiv_q_2exp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_tdiv_q_ui. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_tdiv_qr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_tdiv_qr_ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_tdiv_r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_tdiv_r_2exp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 mpz_tdiv_r_ui. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_tdiv_ui. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 mpz_tstbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 mpz_ui_kronecker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 mpz_ui_pow_ui. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 mpz_ui_sub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 mpz_urandomb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 mpz_urandomm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 mpz_xor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 msqrt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 msub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 mtox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 mult . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

O

operator% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 operator/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 operator<< . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 operator>> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67, 71

P

pow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

R

reallocate_function . . . . . . . . . . . . . . . . . . . . . . . . . . 78 rpow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

S

sdiv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 sgn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70, 71, 72 sqrt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70, 72

T

trunc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

X

xtom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76