GNU MP
The GNU Multiple Precision Arithmetic Library
Edition 3.1.1
18 September 2000
by [email protected]"orn Granlund, Swox AB
Table of Contents
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 that accompany the source code.
GNU MP is a portable library written in C for arbitrary precision arithmetic
on integers, rational numbers, and floatingpoint 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 and Athlon,
Hitachi SuperH and SH2,
HPPA 1.0, 1.1 and 2.0,
Intel Pentium, Pentium Pro/Pentium II, generic x86,
Intel 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 Clipper, IBM ROMP (RT), and Pyramid AP/XP.
There is a mailing list for GMP users. To join it, send a mail to
[email protected] with the word `subscribe' in the message
body (not in the subject line).
For uptodate information on GMP, please see the GMP Home Pages at
http://www.swox.com/gmp/.
Everyone should read section GMP Basics. If you need to install the library
yourself, you need to read section Installing GMP, too.
The rest of the manual can be used for later reference, although it is
probably a good idea to glance through it.
GMP has an autoconf/automake/libtool based configuration system. On a
Unixlike system a basic build can be done with
./configure
make
Some selftests 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 [email protected].
(See section Reporting Bugs, for information on what to include in useful bug
reports.)
All the usual autoconf configure options are available, run `./configure
help' for a summary.
 NonUnix Systems

`configure' needs various Unixlike tools installed. On an MSDOS system
cygwin or djgpp should work. 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.
 Object Directory

To compile in a separate object directory,
cd
to that directory, and
prefix the configure command with the path to the GMP source directory. For
example `../src/gmp/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 from a
separate object directory. Use GNU make
instead.
 @option{disableshared, @option{disablestatic}}

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

The build target can be specified in the usual way, for either native or cross
compilation.
If @option{target} isn't given, `./configure' builds for the host
system as determined by `./config.guess'. On some systems this can't
distinguish between different CPUs in a family, and you should check the
guess. Running `./config.guess' on the target system will also show the
relevant `VENDOROS', if you don't already know what it should be.
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 CPU targets have specific assembly code support. See
`configure.in' for which `mpn' subdirectories get used by each.

Alpha:
`alpha',
`alphaev5',
`alphaev6'

Hitachi:
`sh',
`sh2'

HPPA:
`hppa1.0',
`hppa1.1',
`hppa2.0',
`hppa2.0w'

MIPS:
`mips',
`mips3',

Motorola:
`m68000',
`m68k',
`m88k',
`m88110'

POWER:
`power1',
`power2',
`power2sc',
`powerpc',
`powerpc64'

SPARC:
`sparc',
`sparcv8',
`microsparc',
`supersparc',
`sparcv9',
`ultrasparc',
`sparc64'

80x86 family:
`i386',
`i486',
`i586',
`pentium',
`pentiummmx',
`pentiumpro',
`pentium2',
`pentium3',
`k6',
`k62',
`k63',
`athlon'

Other:
`a29k',
`arm',
`clipper',
`i960',
`ns32k',
`pyramid',
`vax',
`z8k'
CPUs not listed use generic C code. If some of the assembly code causes
problems, the generic C code can be selected with CPU `none'.
 @option{CC, @option{CFLAGS}}

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 configurations specific compiler flags are set based on the target
CPU and compiler, see `CFLAGS' in the generated `Makefile's. 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.
Note that if `CC' is set then `CFLAGS' must also be set. This
applies even if `CC' is merely one of the choices GMP would make itself.
This may change in a future release.
 @option{disablealloca}

By default, GMP allocates temporary workspace using
alloca
if that
function is available, or malloc
if not. If you're working with large
numbers and alloca
overflows the available stack space, you can build
with @option{disablealloca} to use malloc
instead. malloc
will probably be slightly slower than alloca
.
When not using alloca
, it's actually the allocation function
selected with mp_set_memory_functions
that's used, this being
malloc
by default. See section Custom Allocation.
Depending on your system, the only indication of stack overflow might be a
segmentation violation. It might be possible to increase available stack
space with limit
, ulimit
or setrlimit
, or under
DJGPP with stubedit
or _stklen
.
 @option{enablefft}

By default multiplications are done using Karatsuba and 3way ToomCook
algorithms, but a Fermat FFT can be enabled, for use on large to very large
operands. Currently the FFT is recommended only for knowledgeable users who
check the algorithm thresholds for their CPU.
 @option{enablempbsd}

The Berkeley MP compatibility library (`libmp.a') and header file
(`mp.h') are built and installed only if @option{enablempbsd} is used.
See section Berkeley MP Compatible Functions.
 @option{MPN_PATH}

Various assembler versions of mpn subroutines are provided, and, for a given
CPU target, a search is made though a path to choose a version of each. For
example `sparcv8' has path `"sparc32/v8 sparc32 generic"', which
means it looks first for v8 code, falls back on plain sparc32, and finally
falls back on generic C. Knowledgeable users with special requirements can
specify a path with `MPN_PATH="dir list"'. This will normally be
unnecessary because all sensible paths should be available under one or other
CPU target.
 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' and then `./pexpr 68^975+10'.
 Documentation

The document you're now reading is `gmp.texi'. The usual automake
targets are available to make `gmp.ps' and/or `gmp.dvi'. Some
supplementary notes can be found in the `doc' subdirectory.
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 64bit ISA CPUs have both a 64bit ABI and a 32bit ABI defined, the
latter for compatibility with older CPUs in the family. GMP chooses the best
ABI available for a given target system, and this generally gives
significantly greater speed.
The burden is on application programs and cooperating libraries to ensure they
match the ABI chosen by GMP. Fortunately this presents a difficulty only on a
few systems, and if you have one of them then the performance gains are enough
to make it worth the trouble.
Some of what's described in this section may change in future releases of GMP.
 HPPA 2.0

CPU target `hppa2.0' uses the hppa2.0n 32bit ABI, but either a 32bit or
64bit limb.
A 64bit limb is available on HPUX 10 or up when using
c89
. No
gcc
support is planned for 64bit operations in this ABI.
Applications must be compiled with the same options as GMP, which means
c89 +DA2.0 +e D_LONG_LONG_LIMB
A 32bit limb is used in other cases, and no special compiler options are
needed.
CPU target `hppa2.0w' uses the hppa2.0w 64bit ABI, which is available on
HPUX 11 or up when using c89
. gcc
support for this is in
progress. Applications must be compiled for the same ABI, which means
c89 +DD64
 MIPS 3 and 4 under IRIX 6

Targets `mips**irix6*' use the n32 ABI and a 64bit limb. Applications
must be compiled for the same ABI, which means either
gcc mabi=n32
cc n32
 PowerPC 64

CPU target `powerpc64' uses either the 32bit ABI or the AIX 64bit ABI.
The latter is used on targets `powerpc64*aix*' and applications must be
compiled using either
gcc maix64
xlc q64
On other systems the 32bit ABI is used, but with 64bit limbs provided by
long long
in gcc
. Applications must be compiled using
gcc D_LONG_LONG_LIMB
 Sparc V9

On a sparc v9 CPU, either the v8plus 32bit ABI or v9 64bit ABI is used.
Targets `ultrasparc**solaris2.[79]', `sparcv9*solaris2.[79]'
and `sparc64*linux*' use the v9 ABI, if the compiler supports it.
Other targets use the v8plus ABI (but with as much of the v9 ISA as possible
in the circumstances). Note that Solaris prior to 2.7 doesn't save all
registers properly, and hence uses the v8plus ABI.
For the v8plus ABI, applications can be compiled with either
gcc mv8plus
cc xarch=v8plus
For the v9 ABI, applications must be compiled with either
gcc m64 mptr64 Wa,xarch=v9 mcpu=v9
cc xarch=v9
Don't be confused by the names of these options, they're called `arch'
but they effectively control the ABI.
GMP should present no great difficulties for packaging in a binary
distribution.
Libtool is used to build the library and `versioninfo' is set
appropriately, having started from `3:0:0' in GMP 3.0. The GMP 3 series
will be upwardly binary compatible in each release, but may be adding
additional function interfaces. 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 minor version of
GMP.
When building a package for a CPU family, care should be taken to use
`target' 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 `target' in a build so `./config.guess' will detect
the CPU. But a way to manually specify a `target' will be wanted for
systems where `./config.guess' is inexact.
 AIX 4.3

Targets `**aix4.[39]*' have shared libraries disabled since they seem
to fail on AIX 4.3.
 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 @env{PATH}. The bug
is fixed in OpenBSD 2.7, so either upgrade or use GNU m4.
 Sparc V8

Using CPU target `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 Pentium and PentiumPro

The Intel Pentium P5 code is good for its intended P5, but quite slow when run
on Intel P6 class chips (PPro, PII, PIII). `i386' is a better choice
if you're making binaries that must run on both.
 x86 MMX and old GAS

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).
If the target CPU has MMX code but the assembler doesn't support it, a warning
is given and nonMMX 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.
 x86 GCC 2.95.2 `march=pentiumpro'

GCC 2.95.2 miscompiles `mpz/powm.c' when `march=pentiumpro' is
used, so that option is omitted from the @env{CFLAGS} chosen for relevant
CPUs. The problem is believed to be fixed in GCC 2.96.
You might find more uptodate information at http://www.swox.com/gmp/.
 Generic C on a 64bit system

When making a generic C build using `target=none' on a 64bit system
(meaning where
unsigned long
is 64 bits), BITS_PER_MP_LIMB
,
BITS_PER_LONGINT
and BYTES_PER_MP_LIMB
in
`mpn/generic/gmpmparam.h' need to be changed to 64 and 8. This will
hopefully be automated in a future version of GMP.
 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 before you compile GMP. (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.
 Stripped Libraries

GNU binutils `strip' should not be used on the static libraries
`libgmp.a' and `libmp.a', neither directly nor via `make
installstrip'. It can be used on the shared libraries `libgmp.so' and
`libmp.so' though.
Currently (binutils 2.10.0), `strip' extracts archives into a single
directory, 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.
 SunOS 4 Native Tools

The setting for
GSYM_PREFIX
in `config.m4' may be incorrectly
determined when using the native grep
, leading at linktime to
undefined symbols like ___gmpn_add_n
. To fix this, after running
`./configure', change the relevant line in `config.m4' to
`define(<GSYM_PREFIX>, <_>)'.
The ranlib
command will need to be run manually when building a
static library with the native ar
. After `make', run
`ranlib .libs/libgmp.a', and when using @option{enablempbsd} run
`ranlib .libs/libmp.a' too.
 VAX running Ultrix

You need to build and install the GNU assembler before you compile GMP. The VAX
assembly in GMP uses an instruction (
jsobgtr
) that cannot be assembled by
the Ultrix assembler.
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.
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.
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 multiprecision number that fits in a single
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 contains
32 or 64 bits. The C data type for a limb is mp_limb_t
.
There are six classes of functions in the GMP library:

Functions for signed integer arithmetic, with names beginning with
mpz_
. The associated type is mpz_t
. There are about 100
functions in this class.

Functions for rational number arithmetic, with names beginning with
mpq_
. The associated type is mpq_t
. There are about 20
functions in this class, but the functions in the previous class can be used
for performing arithmetic on the numerator and denominator separately.

Functions for floatingpoint arithmetic, with names beginning with
mpf_
. The associated type is mpf_t
. There are about 50
functions is this class.

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

Fast lowlevel 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 timecritical user programs. These functions' names begin with
mpn_
. There are about 30 (hardtouse) functions in this class.
The associated type is array of mp_limb_t
.

Miscellaneous functions. Functions for setting up custom allocation and
functions for generating random numbers.
As a general rule, all GMP functions expect output arguments before input
arguments. This notation is based on an analogy with the assignment operator.
(The BSD MP compatibility functions disobey this rule, having the output
argument(s) 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 floatingpoint
functions for details.
A variable should only be initialized once, or at least cleared out between
each initialization. After a variable has been initialized, it may be
assigned to any number of times.
For efficiency reasons, avoid initializing and clearing out a GMP variable in
a loop. Instead, initialize it before entering the loop, and clear it out
after the loop has exited.
GMP variables are small, containing only a couple of sizes, and pointers to
allocated data. Once you have initialized a GMP variable, you don't need to
worry about space allocation. All functions in GMP automatically allocate
additional space when a variable does not already have enough. They do not,
however, reduce the space when a smaller value is stored. Most of the time
this policy is best, since it avoids frequent reallocation.
When a variable of type mpz_t
is used as a function parameter, it's
effectively a callbyreference, meaning anything the function does to it will
be be done to the original in the caller. When a function is going to return
an mpz_t
result, it should provide a separate parameter or parameters
that it sets, like the GMP library functions do. A return
of an
mpz_t
doesn't return the object, only a pointer to it, and this is
almost certainly not what you want. All this applies to mpq_t
and
mpf_t
too.
Here's an example function accepting an mpz_t
parameter, doing a
certain calculation, and returning a result.
void
myfunction (mpz_t result, 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);
myfunction (r, n, 20L);
mpz_out_str (stdout, 10, r); printf ("\n");
return 0;
}
This example will work if result
and param
are the same
variable, just like the library functions. But sometimes this is tricky to
arrange, and an application might not want to bother for its own subroutines.
mpz_t
is actually implemented as a oneelement array of a certain
structure type. This is why using it to declare a variable gives an object
with the fields GMP needs, but then using it as a parameter passes a pointer
to the object. Note that the actual contents of an mpz_t
are for
internal use only and you should not access them directly if you want your
code to be compatible with future GMP releases.
The GMP code is reentrant and threadsafe, with some exceptions:

The function
mpf_set_default_prec
saves the selected precision in
a global variable.

The function
mp_set_memory_functions
uses several global
variables for storing 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, GMP will not be reentrant either.

The old random number functions (
mpz_random
, etc) use a random number
generator from the C library, usually mrand48
or random
. These
routines are not reentrant, since they rely on global state.
(However the newer random number functions that accept a
gmp_randstate_t
parameter are reentrant.)

If
alloca
is not available, or GMP is configured with
`disablealloca', the library is not reentrant, due to the current
implementation of `stackalloc.c'. In the generated `config.h',
USE_STACK_ALLOC
set to 1 will mean not reentrant.
 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.
This version of GMP is upwardly binary compatible with versions 3.0 and 3.0.1,
and upwardly compatible at the source level with versions 2.0, 2.0.1, and
2.0.2, 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 has reverted to the 2.0.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 3. Please
see the GMP 2 manual for details.
The latest version of the GMP library is available at
ftp://ftp.gnu.org/pub/gnu/gmp. Many sites around the world mirror
`ftp.gnu.org'; please use a mirror site near you, see
http://www.gnu.org/order/ftp.html.
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, you may
want to check http://www.swox.com/gmp/ for patches for this release.
Please include the following in any report,

The GMP version number, and if prepackaged 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
strace
s.

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'.

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'.
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: [email protected].
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.
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
.
The functions for integer arithmetic assume that all integer objects are
initialized. You do that by calling the function mpz_init
.
 Function: void mpz_init (mpz_t integer)

Initialize integer with limb space and set the initial numeric value to
0. Each variable should normally only be initialized once, or at least cleared
out (using
mpz_clear
) between each initialization.
Here is an example of using mpz_init
:
{
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_clear (mpz_t integer)

Free the limb space occupied by integer. Make sure to call this
function for all
mpz_t
variables when you are done with them.
 Function: void * _mpz_realloc (mpz_t integer, mp_size_t new_alloc)

Change the limb space allocation to new_alloc limbs. This function is
not normally called from user code, but it can be used to give memory back to
the heap, or to increase the space of a variable to avoid repeated automatic
reallocation.
 Function: void mpz_array_init (mpz_t integer_array[], size_t array_size, mp_size_t fixed_num_bits)

Allocate fixed limb space for all array_size integers in
integer_array. The fixed allocation for each integer in the array is
enough to store fixed_num_bits. If the fixed space will be insufficient
for storing the result of a subsequent calculation, the result is
unpredictable.
This function is useful for decreasing the working set for some algorithms
that use large integer arrays.
There is no way to deallocate the storage allocated by this function.
Don't call mpz_clear
!
These functions assign new values to already initialized integers
(see section Initialization Functions).
 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.
 Function: int mpz_set_str (mpz_t rop, char *str, int base)

Set the value of rop from str, a '\0'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 up to the '\0' is a valid
number in base base. Otherwise it returns 1.
[It turns out that it is not entirely true that this function ignores
whitespace. 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 whitespace 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?]
 Function: void mpz_swap (mpz_t rop1, mpz_t rop2)

Swap the values rop1 and rop2 efficiently.
For convenience, GMP provides a parallel series of initializeandset 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 initializeandset 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.)
This section describes functions for converting GMP integers to standard C
types. Functions for converting to GMP integers are described in
section Assignment Functions and section Input and Output Functions.
 Function: mp_limb_t mpz_getlimbn (mpz_t op, mp_size_t n)

Return limb #n from op. This function allows for very efficient
decomposition of a number in its limbs.
The function mpz_size
can be used to determine the useful range for
n.
 Function: unsigned long int mpz_get_ui (mpz_t op)

Return the least significant part from op. This function combined with
mpz_tdiv_q_2exp(..., op, CHAR_BIT*sizeof(unsigned long
int))
can be used to decompose an integer into unsigned longs.
 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 large 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: 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
, space for the result string is allocated using the
default allocation function.
If str is not NULL
, it should point to a block of storage enough large
for the result. To find out the right amount of space to provide for
str, use mpz_sizeinbase (op, base) + 2
. The two
extra bytes are for a possible minus sign, and for the terminating null
character.
A pointer to the result string is returned. This pointer will will either
equal str, or if that is NULL
, will point to the allocated storage.
 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)

@ifnottex
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)

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)

@ifnottex
Set rop to op1 times op2.
 Function: void mpz_addmul_ui (mpz_t rop, mpz_t op1, unsigned long int op2)

@ifnottex
Add op1 times op2 to rop.
 Function: void mpz_mul_2exp (mpz_t rop, mpz_t op1, unsigned long int op2)

@ifnottex
Set rop to op1 times 2 raised to op2. This operation can
also be defined as a left shift, op2 steps.
 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.
Division is undefined if the divisor is zero, and passing a zero divisor to the
divide or modulo functions, as well passing a zero mod argument to the
mpz_powm
and mpz_powm_ui
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.
There are three main groups of division functions:

Functions that truncate the quotient towards 0. The names of these functions
start with
mpz_tdiv
. The `t' in the name is short for
`truncate'.

Functions that round the quotient towards
@ifnottex
infinity).
The names of these routines start with
mpz_fdiv
. The `f' in the
name is short for `floor'.

Functions that round the quotient towards
@ifnottex
+infinity.
The names of these routines start with
mpz_cdiv
. The `c' in the
name is short for `ceil'.
For each rounding mode, there are a couple of variants. Here `q' means
that the quotient is computed, while `r' means that the remainder is
computed. Functions that compute both the quotient and remainder have
`qr' in the name.
 Function: void mpz_tdiv_q (mpz_t q, mpz_t n, mpz_t d)

 Function: unsigned long int mpz_tdiv_q_ui (mpz_t q, mpz_t n, unsigned long int d)

Set q to [n/d], truncated towards 0.
The function mpz_tdiv_q_ui
returns the absolute value of the true
remainder.
 Function: void mpz_tdiv_r (mpz_t r, mpz_t n, mpz_t d)

 Function: unsigned long int mpz_tdiv_r_ui (mpz_t r, mpz_t n, unsigned long int d)

@ifnottex
Set r to (n  [n/d] * d), where the quotient is
truncated towards 0. Unless r becomes zero, it will get the same sign as
n.
The function mpz_tdiv_r_ui
returns the absolute value of the remainder.
 Function: void mpz_tdiv_qr (mpz_t q, mpz_t r, mpz_t n, mpz_t d)

 Function: unsigned long int mpz_tdiv_qr_ui (mpz_t q, mpz_t r, mpz_t n, unsigned long int d)

@ifnottex
Set q to [n/d], truncated towards 0. Set r to (n
 [n/d] * d). Unless r becomes zero, it will get the
same sign as n. If q and r are the same variable, the
results are undefined.
The function mpz_tdiv_qr_ui
returns the absolute value of the remainder.
 Function: unsigned long int mpz_tdiv_ui (mpz_t n, unsigned long int d)

Like
mpz_tdiv_r_ui
, but the remainder is not stored anywhere; its
absolute value is just returned.
 Function: void mpz_fdiv_q (mpz_t q, mpz_t n, mpz_t d)

 Function: unsigned long int mpz_fdiv_q_ui (mpz_t q, mpz_t n, unsigned long int d)

@ifnottex
Set q to n/d, rounded towards infinity.
The function mpz_fdiv_q_ui
returns the remainder.
 Function: void mpz_fdiv_r (mpz_t r, mpz_t n, mpz_t d)

 Function: unsigned long int mpz_fdiv_r_ui (mpz_t r, mpz_t n, unsigned long int d)

@ifnottex
Set r to (n  n/d * d), where the quotient is
rounded towards infinity. Unless r becomes zero, it will get the
same sign as d.
The function mpz_fdiv_r_ui
returns the remainder.
 Function: void mpz_fdiv_qr (mpz_t q, mpz_t r, mpz_t n, mpz_t d)

 Function: unsigned long int mpz_fdiv_qr_ui (mpz_t q, mpz_t r, mpz_t n, unsigned long int d)

@ifnottex
Set q to n/d, rounded towards infinity. Set r
to (n  n/d * d). Unless r becomes zero, it
will get the same sign as d. If q and r are the same
variable, the results are undefined.
The function mpz_fdiv_qr_ui
returns the remainder.
 Function: unsigned long int mpz_fdiv_ui (mpz_t n, unsigned long int d)

Like
mpz_fdiv_r_ui
, but the remainder is not stored anywhere; it is just
returned.
 Function: void mpz_cdiv_q (mpz_t q, mpz_t n, mpz_t d)

 Function: unsigned long int mpz_cdiv_q_ui (mpz_t q, mpz_t n, unsigned long int d)

@ifnottex
Set q to n/d, rounded towards +infinity.
The function mpz_cdiv_q_ui
returns the negated remainder.
 Function: void mpz_cdiv_r (mpz_t r, mpz_t n, mpz_t d)

 Function: unsigned long int mpz_cdiv_r_ui (mpz_t r, mpz_t n, unsigned long int d)

@ifnottex
Set r to (n  n/d * d), where the quotient is
rounded towards +infinity. Unless r becomes zero, it will get the
opposite sign as d.
The function mpz_cdiv_r_ui
returns the negated remainder.
 Function: void mpz_cdiv_qr (mpz_t q, mpz_t r, mpz_t n, mpz_t d)

 Function: unsigned long int mpz_cdiv_qr_ui (mpz_t q, mpz_t r, mpz_t n, unsigned long int d)

@ifnottex
Set q to n/d, rounded towards +infinity. Set r
to (n  n/d * d). Unless r becomes zero, it
will get the opposite sign as d. If q and r are the same
variable, the results are undefined.
The function mpz_cdiv_qr_ui
returns the negated remainder.
 Function: unsigned long int mpz_cdiv_ui (mpz_t n, unsigned long int d)

Like
mpz_tdiv_r_ui
, but the remainder is not stored anywhere; its
negated value is just returned.
 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 nonnegative.
The function mpz_mod_ui
returns the remainder.
 Function: void mpz_divexact (mpz_t q, mpz_t n, mpz_t d)

Set q to n/d. This function produces correct results only
when it is known in advance that d divides n.
Since mpz_divexact is much faster than any of the other routines that produce
the quotient (see section References Jebelean), it is the best choice for instances
in which exact division is known to occur, such as reducing a rational to
lowest terms.
 Function: void mpz_tdiv_q_2exp (mpz_t q, mpz_t n, unsigned long int d)

@ifnottex
Set q to n divided by 2 raised to d. The quotient is truncated
towards 0.
 Function: void mpz_tdiv_r_2exp (mpz_t r, mpz_t n, unsigned long int d)

@ifnottex
Divide n by (2 raised to d), rounding the quotient towards 0, and
put the remainder in r.
Unless it is zero, r will have the same sign as n.
 Function: void mpz_fdiv_q_2exp (mpz_t q, mpz_t n, unsigned long int d)

@ifnottex
Set q to n divided by 2 raised to d, rounded towards
infinity.
This operation can also be defined as arithmetic right shift d bit
positions.
 Function: void mpz_fdiv_r_2exp (mpz_t r, mpz_t n, unsigned long int d)

@ifnottex
Divide n by (2 raised to d), rounding the quotient towards
infinity, and put the remainder in r.
The sign of r will always be positive.
This operation can also be defined as masking of the d least significant
bits.
 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)

@ifnottex
Set rop to (base raised to exp)
mod
mod. If
exp is negative, the result is undefined.
 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)

@ifnottex
Set rop to base raised to exp. The case of 0^0 yields 1.
 Function: int mpz_root (mpz_t rop, mpz_t op, unsigned long int n)

@ifnottex
Set rop to the truncated integer part of the nth root of op.
Return nonzero 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)

@ifnottex
Set rop to the truncated integer part of the square root of op.
 Function: void mpz_sqrtrem (mpz_t rop1, mpz_t rop2, mpz_t op)

@ifnottex
Set rop1 to the truncated integer part of the square root of op,
like
mpz_sqrt
. Set rop2 to
oprop1*rop1,
(i.e., 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)

@ifnottex
Return nonzero if op is a perfect power, i.e., if there exist integers
a and b, with b > 1, such that op equals a raised to
b. Return zero otherwise.
 Function: int mpz_perfect_square_p (mpz_t op)

Return nonzero if op is a perfect square, i.e., if the square root of
op is an integer. Return zero otherwise.
 Function: int mpz_probab_prime_p (mpz_t n, int reps)

If this function returns 0, n is definitely not prime. If it
returns 1, then n is `probably' prime. If it returns 2, then
n is surely prime. Reasonable values of reps vary from 5 to 10; a
higher value lowers the probability for a nonprime to pass as a
`probable' prime.
The function uses MillerRabin's probabilistic test.
 Function: int 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, but for for
practical purposes it's adequate, since 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 either of 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 nonzero.
 Function: void mpz_gcdext (mpz_t g, mpz_t s, mpz_t t, mpz_t a, mpz_t b)

Compute g, s, and t, such that as +
bt = g =
gcd
(a, b). If t is
NULL
, that argument is not computed.
 Function: void mpz_lcm (mpz_t rop, mpz_t op1, mpz_t op2)

Set rop to the least common multiple of op1 and op2.
 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. Return nonzero if an inverse exists, zero otherwise. When the
function returns zero, rop is undefined.
 Function: int mpz_jacobi (mpz_t op1, mpz_t op2)

 Function: int mpz_legendre (mpz_t op1, mpz_t op2)

Compute the Jacobi and Legendre symbols, respectively. op2 should be
odd and must be positive.
 Function: int mpz_si_kronecker (long a, mpz_t b);

 Function: int mpz_ui_kronecker (unsigned long 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);

@ifnottex
Calculate the value of the Kronecker/Jacobi symbol (a/b), with the
Kronecker extension (a/2)=(2/a) when a odd, or (a/2)=0 when a even.
All values of a and b give a welldefined result. See Henri
Cohen, section 1.4.2, for more information (see section References). 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
@ifnottex
n over k
and store the result in rop. Negative values of n are supported
by
mpz_bin_ui
, using the identity
@ifnottex
bin(n,k) = (1)^k * bin(n+k1,k)
(see Knuth volume 1 section 1.2.6 part G).
 Function: void mpz_fib_ui (mpz_t rop, unsigned long int n)

Compute the nth Fibonacci number and store the result in rop.
 Function: int mpz_cmp (mpz_t op1, mpz_t op2)

@ifnottex
Compare op1 and op2. Return a positive value if op1 >
op2, zero if op1 = op2, and a negative value if op1 <
op2.
 Macro: int mpz_cmp_ui (mpz_t op1, unsigned long int op2)

 Macro: int mpz_cmp_si (mpz_t op1, signed long int op2)

@ifnottex
Compare op1 and op2. Return a positive value if op1 >
op2, zero if op1 = op2, and a negative value if op1 <
op2.
These functions are actually implemented as macros. They evaluate their
arguments multiple times.
 Function: int mpz_cmpabs (mpz_t op1, mpz_t op2)

 Function: int mpz_cmpabs_ui (mpz_t op1, unsigned long int op2)

@ifnottex
Compare the absolute values of op1 and op2. Return a positive
value if op1 > op2, zero if op1 = op2, and a negative
value if op1 < op2.
 Macro: int mpz_sgn (mpz_t op)

@ifnottex
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.
These functions behave as if two's complement arithmetic were used (although
signmagnitude is used by the actual implementation).
 Function: void mpz_and (mpz_t rop, mpz_t op1, mpz_t op2)

Set rop to op1 logicaland op2.
 Function: void mpz_ior (mpz_t rop, mpz_t op1, mpz_t op2)

Set rop to op1 inclusiveor op2.
 Function: void mpz_xor (mpz_t rop, mpz_t op1, mpz_t op2)

Set rop to op1 exclusiveor 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)

For nonnegative numbers, return the population count of op. For
negative numbers, return the largest possible value (MAX_ULONG).
 Function: unsigned long int mpz_hamdist (mpz_t op1, mpz_t op2)

If op1 and op2 are both nonnegative, return the hamming distance
between the two operands. Otherwise, return the largest possible value
(MAX_ULONG).
It is possible to extend this function to return a useful value when the
operands are both negative, but the current implementation returns
MAX_ULONG in this case. Do not depend on this behavior, since
it will change in a future release.
 Function: unsigned long int mpz_scan0 (mpz_t op, unsigned long int starting_bit)

Scan op, starting with bit starting_bit, towards more significant
bits, until the first clear bit is found. Return the index of the found bit.
 Function: unsigned long int mpz_scan1 (mpz_t op, unsigned long int starting_bit)

Scan op, starting with bit starting_bit, towards more significant
bits, until the first set bit is found. Return the index of the found bit.
 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)

Check bit bit_index in op and return 0 or 1 accordingly.
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 whitespace 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 bigendian).
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 32bit and 64bit 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 32bit and 64bit
machines.
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 section Random Number Functions 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
@ifnottex
0 to 2^n  1,
inclusive.
The variable state must be initialized by calling one of the
gmp_randinit
functions (section Random State Initialization) 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
@ifnottex
n  1, inclusive.
The variable state must be initialized by calling one of the
gmp_randinit
functions (section Random State Initialization)
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 cornercase bugs. The random number will be in the range
@ifnottex
0 to 2^n  1,
inclusive.
The variable state must be initialized by calling one of the
gmp_randinit
functions (section Random State Initialization)
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 cornercase bugs. Negative random numbers are generated
when max_size is negative.
This function is obsolete. Use mpz_rrandomb
instead.
 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 nonzero 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 nonzero if
yes, zero if no. These macros evaluate their arguments 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 returned value will be exact or 1 too
big. If base is a power of 2, the returned value will always be exact.
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 terminating '\0').
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. Note that
this is an incompatible change from version 1 of the library.
 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.
 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: void mpq_swap (mpq_t rop1, mpq_t rop2)

Swap the values rop1 and rop2 efficiently.
 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)

@ifnottex
Set product to multiplier times multiplicand.
 Function: void mpq_div (mpq_t quotient, mpq_t dividend, mpq_t divisor)

Set quotient to dividend/divisor.
 Function: void mpq_neg (mpq_t negated_operand, mpq_t operand)

Set negated_operand to operand.
 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.
 Function: int mpq_cmp (mpq_t op1, mpq_t op2)

@ifnottex
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)

@ifnottex
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.
This routine allows that num2 and den2 have common factors.
This function is actually implemented as a macro. It evaluates its
arguments multiple times.
 Macro: int mpq_sgn (mpq_t op)

@ifnottex
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 nonzero if op1 and op2 are equal, zero if they are
nonequal. Although
mpq_cmp
can be used for the same purpose, this
function is much faster.
The set of mpq
functions is quite small. In particular, there are few
functions for either input or output. But there are two macros that allow us
to apply any mpz
function on the numerator or denominator of a rational
number. If these macros are used to assign to the rational number,
mpq_canonicalize
normally need to be called afterwards.
 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.
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 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: double mpq_get_d (mpq_t op)

Convert op to a double.
 Function: void mpq_set_d (mpq_t rop, double d)

Set rop to the value of d, without rounding.
These functions assign between either the numerator or denominator of a
rational, and an integer. Instead of using these functions, it is preferable
to use the more general mechanisms mpq_numref
and mpq_denref
,
together with mpz_set
.
 Function: void mpq_set_num (mpq_t rational, mpz_t numerator)

Copy numerator to the numerator of rational. When this risks to
make the numerator and denominator of rational have common factors, you
have to pass rational to
mpq_canonicalize
before any operations
are performed on rational.
This function is equivalent to
mpz_set (mpq_numref (rational), numerator)
.
 Function: void mpq_set_den (mpq_t rational, mpz_t denominator)

Copy denominator to the denominator of rational. When this risks
to make the numerator and denominator of rational have common factors,
or if the denominator might be negative, you have to pass rational to
mpq_canonicalize
before any operations are performed on rational.
In version 1 of the library, negative denominators were handled by
copying the sign to the numerator. That is no longer done.
This function is equivalent to
mpz_set (mpq_denref (rational), denominators)
.
 Function: void mpq_get_num (mpz_t numerator, mpq_t rational)

Copy the numerator of rational to the integer numerator, to
prepare for integer operations on the numerator.
This function is equivalent to
mpz_set (numerator, mpq_numref (rational))
.
 Function: void mpq_get_den (mpz_t denominator, mpq_t rational)

Copy the denominator of rational to the integer denominator, to
prepare for integer operations on the denominator.
This function is equivalent to
mpz_set (denominator, mpq_denref (rational))
.
This chapter describes the GMP functions for performing floating point
arithmetic. These functions start with the prefix mpf_
.
GMP floating point numbers are stored in objects of type mpf_t
.
The GMP floatingpoint functions have an interface that is similar to the GMP
integer functions. The function prefix for floatingpoint operations is
mpf_
.
There is one significant characteristic of floatingpoint numbers that has
motivated a difference between this function class and other GMP function
classes: the inherent inexactness of floating point arithmetic. The user has
to specify the precision of each variable. A computation that assigns a
variable will take place with the precision of the assigned variable; the
precision of variables used as input is ignored.
The precision of a calculation is defined as follows: Compute the requested
operation exactly (with "infinite precision"), and truncate the result to
the destination variable precision. Even if the user has asked for a very
high precision, GMP will not calculate with superfluous digits. For example,
if two lowprecision numbers of nearly equal magnitude are added, the
precision of the result will be limited to what is required to represent the
result accurately.
The GMP floatingpoint functions are not intended as a smooth extension
to the IEEE P754 arithmetic. Specifically, the results obtained on one
computer often differs from the results obtained on a computer with a
different word size.
 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.
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 floatingpoint 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 NewtonRaphson, making the computation precision
closely match the actual accurate part of the numbers.
 Function: void mpf_set_prec (mpf_t rop, unsigned long int prec)

Set the precision of rop to be at least prec bits.
Since changing the precision involves calls to
realloc
, this routine
should not be called in a tight loop.
 Function: unsigned long int mpf_get_prec (mpf_t op)

Return the precision actually used for assignments of op.
 Function: void mpf_set_prec_raw (mpf_t rop, unsigned long int prec)

Set the precision of rop to be at least prec bits. This
is a lowlevel function that does not change the allocation. The prec
argument must not be larger that the precision previously returned by
mpf_get_prec
. It is crucial that the precision of rop is
ultimately reset to exactly the value returned by mpf_get_prec
before
the first call to mpf_set_prec_raw
.
These functions assign new values to already initialized floats
(see section Initialization Functions).
 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 `[email protected]' 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 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 whitespace 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 up to the '\0' is a valid number
in base base. Otherwise it returns 1.
 Function: void mpf_swap (mpf_t rop1, mpf_t rop2)

Swap the values rop1 and rop2 efficiently.
For convenience, GMP provides a parallel series of initializeandset 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 initializeandset 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
.
 Function: double mpf_get_d (mpf_t op)

Convert op to a double.
 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. The base may vary
from 2 to 36. Generate at most n_digits significant digits, or if
n_digits is 0, the maximum number of digits accurately representable by
op.
If str is NULL
, space for the mantissa is allocated using the default
allocation function.
If str is not NULL
, it should point to a block of storage enough large
for the mantissa, i.e., n_digits + 2. The two extra bytes are for a
possible minus sign, and for the terminating null character.
The exponent is written through the pointer expptr.
If n_digits is 0, the maximum number of digits meaningfully achievable
from the precision of op will be generated. Note that the space
requirements for str in this case will be impossible for the user to
predetermine. Therefore, you need to pass NULL
for the string argument
whenever n_digits is 0.
The generated string is a fraction, with an implicit radix point immediately
to the left of the first digit. For example, the number 3.1416 would be
returned as "31416" in the string and 1 written at expptr.
A pointer to the result string is returned. This pointer will will either
equal str, or if that is NULL
, will point to the allocated storage.
 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)

@ifnottex
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)

@ifnottex
Set rop to op1 times 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)

@ifnottex
Set rop to the square root of op.
 Function: void mpf_pow_ui (mpf_t rop, mpf_t op1, unsigned long int op2)

@ifnottex
Set rop to op1 raised to the power 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)

@ifnottex
Set rop to op1 times 2 raised to op2.
 Function: void mpf_div_2exp (mpf_t rop, mpf_t op1, unsigned long int op2)

@ifnottex
Set rop to op1 divided by 2 raised to op2.
 Function: int mpf_cmp (mpf_t op1, mpf_t 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)

@ifnottex
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 nonzero if the first op3 bits of op1 and op2 are
equal, zero otherwise. I.e., test of op1 and op2 are
approximately equal.
 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.
 Macro: int mpf_sgn (mpf_t op)

@ifnottex
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.
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)

Output op on stdio stream stream, as a string of digits in
base base. The base may vary from 2 to 36. Print at most
n_digits significant digits, or if n_digits is 0, the maximum
number of digits accurately representable by op.
In addition to the significant digits, a leading `0.' and a
trailing exponent, in the form `eNNN', are printed. If base
is greater than 10, `@' will be used instead of `e' as
exponent delimiter.
Return the number of bytes written, or if an error occurred, return 0.
 Function: size_t mpf_inp_str (mpf_t rop, FILE *stream, int base)

Input a string in base base from stdio stream stream, and put the
read float in rop. The string is of the form `[email protected]' 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 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.
 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: 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 Random State Initialization)
before invoking this function.
 Function: void mpf_random2 (mpf_t rop, mp_size_t max_size, mp_exp_t max_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 cornercase bugs. Negative random numbers
are generated when max_size is negative.
This chapter describes lowlevel GMP functions, used to implement the highlevel
GMP functions, but also intended for timecritical 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,
inplace 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 size limbs.
cy = mpn_add_n (destp, s1p, s2p, size)
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, s1size}.
 Function: mp_limb_t mpn_add_n (mp_limb_t *rp, const mp_limb_t *s1p, const mp_limb_t *s2p, mp_size_t size)

Add {s1p, size} and {s2p, size}, and
write the size least significant limbs of the result to rp.
Return carry, either 0 or 1.
This is the lowestlevel function for addition. It is the preferred function
for addition, since it is written in assembly for most targets. 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 size, mp_limb_t s2limb)

Add {s1p, size} and s2limb, and write the
size 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 s1size, const mp_limb_t *s2p, mp_size_t s2size)

Add {s1p, s1size} and {s2p,
s2size}, and write the s1size least significant limbs of
the result to rp. Return carry, either 0 or 1.
This function requires that s1size is greater than or equal to
s2size.
 Function: mp_limb_t mpn_sub_n (mp_limb_t *rp, const mp_limb_t *s1p, const mp_limb_t *s2p, mp_size_t size)

Subtract {s2p, s2size} from {s1p,
size}, and write the size least significant limbs of the result
to rp. Return borrow, either 0 or 1.
This is the lowestlevel function for subtraction. It is the preferred
function for subtraction, since it is written in assembly for most targets.
 Function: mp_limb_t mpn_sub_1 (mp_limb_t *rp, const mp_limb_t *s1p, mp_size_t size, mp_limb_t s2limb)

Subtract s2limb from {s1p, size}, and write the
size 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 s1size, const mp_limb_t *s2p, mp_size_t s2size)

Subtract {s2p, s2size} from {s1p,
s1size}, and write the s1size least significant limbs of
the result to rp. Return borrow, either 0 or 1.
This function requires that s1size is greater than or equal to
s2size.
 Function: void mpn_mul_n (mp_limb_t *rp, const mp_limb_t *s1p, const mp_limb_t *s2p, mp_size_t size)

Multiply {s1p, size} and {s2p, size},
and write the entire result to rp.
The destination has to have space for 2*size limbs, even if the
significant result might be one limb smaller.
 Function: mp_limb_t mpn_mul_1 (mp_limb_t *rp, const mp_limb_t *s1p, mp_size_t size, mp_limb_t s2limb)

Multiply {s1p, size} and s2limb, and write the
size least significant limbs of the product to rp. Return
the most significant limb of the product.
This is a lowlevel function that is a building block for general
multiplication as well as other operations in GMP. It is written in assembly
for most targets.
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 size, mp_limb_t s2limb)

Multiply {s1p, size} and s2limb, and add the
size least significant limbs of the product to {rp,
size} and write the result to rp. Return
the most significant limb of the product, plus carryout from the addition.
This is a lowlevel function that is a building block for general
multiplication as well as other operations in GMP. It is written in assembly
for most targets.
 Function: mp_limb_t mpn_submul_1 (mp_limb_t *rp, const mp_limb_t *s1p, mp_size_t size, mp_limb_t s2limb)

Multiply {s1p, size} and s2limb, and subtract the
size least significant limbs of the product from {rp,
size} and write the result to rp. Return the most
significant limb of the product, minus borrowout from the subtraction.
This is a lowlevel 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 targets.
 Function: mp_limb_t mpn_mul (mp_limb_t *rp, const mp_limb_t *s1p, mp_size_t s1size, const mp_limb_t *s2p, mp_size_t s2size)

Multiply {s1p, s1size} and {s2p,
s2size}, and write the result to rp. Return the most
significant limb of the result.
The destination has to have space for s1size + s2size
limbs, even if the result might be one limb smaller.
This function requires that s1size is greater than or equal to
s2size. The destination must be distinct from either 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}. Write the quotient
at qp and the remainder at rp.
The quotient written at qp will be nn  dn + 1 limbs.
The remainder written at rp will be dn limbs.
It is required that nn is greater than or equal to dn. The
qxn operand must be zero.
The quotient is rounded towards 0.
No overlap between arguments is permitted.
 Function: mp_limb_t mpn_divrem (mp_limb_t *r1p, mp_size_t xsize, mp_limb_t *rs2p, mp_size_t rs2size, const mp_limb_t *s3p, mp_size_t s3size)

[This function is obsolete. Please call
mpn_tdiv_qr
instead for
best performance.]
Divide {rs2p, rs2size} by {s3p, s3size}, 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 s3size limbs long (i.e., as many limbs as the divisor).
In addition to an integer quotient, xsize fraction limbs are developed,
and stored after the integral limbs. For most usages, xsize will be
zero.
It is required that rs2size is greater than or equal to s3size.
It is required that the most significant bit of the divisor is set.
If the quotient is not needed, pass rs2p + s3size 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 rs2size  s3size +
xsize limbs large.
 Function: mp_limb_t mpn_divrem_1 (mp_limb_t *r1p, mp_size_t xsize, mp_limb_t *s2p, mp_size_t s2size, mp_limb_t s3limb)

 Macro: mp_limb_t mpn_divmod_1 (mp_limb_t *r1p, mp_limb_t *s2p, mp_size_t s2size, mp_limb_t s3limb)

Divide {s2p, s2size} by s3limb, and write the quotient
at r1p. Return the remainder.
The integer quotient is written to {r1p+xsize, s2size} and
in addition xsize fraction limbs are developed and written to
{r1p, xsize}. Either or both s2size and xsize can
be zero. For most usages, xsize will be zero.
mpn_divmod_1
exists for upward source compatibility and is simply a
macro calling mpn_divrem_1
with an xsize 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 rs2size, const mp_limb_t *s3p, mp_size_t s3size)

This interface is obsolete. It will disappear from future releases.
Use
mpn_divrem
in its stead.
 Macro: mp_limb_t mpn_divexact_by3 (mp_limb_t *rp, mp_limb_t *sp, mp_size_t size)

 Function: mp_limb_t mpn_divexact_by3c (mp_limb_t *rp, mp_limb_t *sp, mp_size_t size, mp_limb_t carry)

Divide {sp, size} by 3, expecting it to divide exactly, and
writing the result to {rp, size}. If 3 divides exactly, the
return value is zero and the result is the quotient. If not, the return value
is nonzero 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. mpn_divexact_by3
is simply a macro calling
mpn_divexact_by3c
with a 0 carry parameter.
These routines use a multiplybyinverse 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
@ifnottex
c*b^n + ai = 3*q,
where b is the size of a limb
@ifnottex
(2^32 or 2^64).
c is always 0, 1 or 2, and the initial carry must also be 0, 1 or 2
(these are both borrows really). When c=0, clearly q=(ai)/3.
When
@ifnottex
c!=0, the remainder (ai) mod 3
is given by 3c, because
@ifnottex
b == 1 mod 3.
 Function: mp_limb_t mpn_mod_1 (mp_limb_t *s1p, mp_size_t s1size, mp_limb_t s2limb)

Divide {s1p, s1size} by s2limb, and return the remainder.
s1size can be zero.
 Function: mp_limb_t mpn_preinv_mod_1 (mp_limb_t *s1p, mp_size_t s1size, mp_limb_t s2limb, mp_limb_t s3limb)

This interface is obsolete. It will disappear from future releases.
Use
mpn_mod_1
in its stead.
 Function: mp_limb_t mpn_bdivmod (mp_limb_t *rp, mp_limb_t *s1p, mp_size_t s1size, const mp_limb_t *s2p, mp_size_t s2size, unsigned long int d)

The function puts the low [d/BITS_PER_MP_LIMB] limbs of
q =
{s1p, s1size}/{s2p, s2size}
mod 2^d
at rp,
and returns the high d mod BITS_PER_MP_LIMB bits of q.
{s1p, s1size}  q * {s2p, s2size}
mod 2^(s1size*BITS_PER_MP_LIMB)
is placed at s1p.
Since the low [d/BITS_PER_MP_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 s1size * BITS_PER_MP_LIMB >= D,
and that {s2p, s2size} 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 *src_ptr, mp_size_t src_size, unsigned long int count)

Shift {src_ptr, src_size} count bits to the left, and
write the src_size least significant limbs of the result to
rp. count might be in the range 1 to n  1, on an
nbit machine. The bits shifted out to the left are returned.
Overlapping of the destination space and the source space is allowed in this
function, provided rp >= src_ptr.
This function is written in assembly for most targets.
 Function: mp_limp_t mpn_rshift (mp_limb_t *rp, const mp_limb_t *src_ptr, mp_size_t src_size, unsigned long int count)

Shift {src_ptr, src_size} count bits to the right, and
write the src_size most significant limbs of the result to
rp. count might be in the range 1 to n  1, on an
nbit machine. The bits shifted out to the right are returned.
Overlapping of the destination space and the source space is allowed in this
function, provided rp <= src_ptr.
This function is written in assembly for most targets.
 Function: int mpn_cmp (const mp_limb_t *s1p, const mp_limb_t *s2p, mp_size_t size)

Compare {s1p, size} and {s2p, size} and
return a positive value if s1 > src2, 0 of they are equal, and a negative
value if s1 < src2.
 Function: mp_size_t mpn_gcd (mp_limb_t *rp, mp_limb_t *s1p, mp_size_t s1size, mp_limb_t *s2p, mp_size_t s2size)

Puts at rp the greatest common divisor of {s1p,
s1size} and {s2p, s2size}; both source
operands are destroyed by the operation. The size in limbs of the greatest
common divisor is returned.
{s1p, s1size} must have at least as many bits as
{s2p, s2size}, and {s2p, s2size} must be odd.
 Function: mp_limb_t mpn_gcd_1 (const mp_limb_t *s1p, mp_size_t s1size, mp_limb_t s2limb)

Return the greatest common divisor of {s1p, s1size}
and s2limb, where s2limb (as well as s1size)
must be different from 0.
 Function: mp_size_t mpn_gcdext (mp_limb_t *r1p, mp_limb_t *r2p, mp_size_t *r2size, mp_limb_t *s1p, mp_size_t s1size, mp_limb_t *s2p, mp_size_t s2size)

Compute the greatest common divisor of {s1p, s1size} and
{s2p, s2size}. Store the gcd at r1p and return its size
in limbs. Write the first cofactor at r2p and store its size in
*r2size. If the cofactor is negative, *r2size is negative and
r2p is the absolute value of the cofactor.
{s1p, s1size} must be greater than or equal to {s2p,
s2size}. Neither operand may equal 0. Both source operands are
destroyed, plus one limb past the end of each, ie. {s1p,
s1size+1} and {s2p, s2size+1}.
 Function: mp_size_t mpn_sqrtrem (mp_limb_t *r1p, mp_limb_t *r2p, const mp_limb_t *sp, mp_size_t size)

Compute the square root of {sp, size} and put the result at
r1p. Write the remainder at r2p, unless r2p is
NULL
.
Return the size of the remainder, whether r2p was NULL
or nonNULL
.
Iff the operand was a perfect square, the return value will be 0.
The areas at r1p and sp have to be distinct. The areas at
r2p and sp have to be identical or completely separate, not
partially overlapping.
@ifnottex
The area at r1p needs to have space for ceil(size/2) limbs.
The area at r2p needs to be size limbs large.
 Function: mp_size_t mpn_get_str (unsigned char *str, int base, mp_limb_t *s1p, mp_size_t s1size)

Convert {s1p, s1size} to a raw unsigned char array in base
base. 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. There may be leading zeros in the string.
The area at s1p is clobbered.
Return the number of characters in str.
The area at str has to have space for the largest possible number
represented by a s1size long limb array, plus one extra character.
 Function: mp_size_t mpn_set_str (mp_limb_t *r1p, const char *str, size_t strsize, int base)

Convert the raw unsigned char array at str of length strsize to
a limb array {s1p, s1size}. The base of str is
base.
Return the number of limbs stored in r1p.
 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 r1size)

 Function: void mpn_random2 (mp_limb_t *r1p, mp_size_t r1size)

Generate a random number of length r1size and store it at r1p.
The most significant limb is always nonzero.
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, unsigned long int size)

Count the number of set bits in {s1p, size}.
 Function: unsigned long int mpn_hamdist (const mp_limb_t *s1p, const mp_limb_t *s2p, unsigned long int size)

Compute the hamming distance between {s1p, size} and
{s2p, size}.
 Function: int mpn_perfect_square_p (const mp_limb_t *s1p, mp_size_t size)

Return nonzero iff {s1p, size} is a perfect square.
There are two groups of random number functions in GNU MP; older
functions that call C library random number generators, rely on a global
state, and aren't very random; and newer functions that don't have these
problems. The newer functions are selfcontained, they accept a random
state parameter that supplants global state, and generate good random
numbers.
The random state parameter is of the type gmp_randstate_t
. It must be
initialized by a call to one of the gmp_randinit
functions (section Random State Initialization). The initial seed is set using one of the
gmp_randseed
functions (section Random State Initialization).
The size of the seed determines the number of different sequences of
random numbers that is possible to generate. The "quality" of the
seed is the randomness of a given seed compared to the previous seed
used and affects the randomness of separate number sequences.
The algorithm for assigning seed is critical if the generated random numbers
are to be used for important applications, such as generating cryptographic
keys.
The traditional method is to use the current system time for seeding. One has
to be careful when using the current time though. If the application seeds the
random functions very often, say several times per second, and the resolution
of the system clock is comparatively low, like one second, the same sequence of
numbers will be generated until the system clock ticks. Furthermore, the
current system time is quite easy to guess, so a system depending on any
unpredictability of the random number sequence should absolutely not use that
as its only source for a seed value.
On some systems there is a special device, often called /dev/random
,
which provides a source of somewhat random numbers more usable as seed.
The functions actually generating random functions are documented under
"Miscellaneous Functions" in their respective function class:
section Miscellaneous Functions, section Miscellaneous Functions.
See section Random Number Functions for a discussion on how to choose the
initial seed value passed to these functions.
 Function: void gmp_randinit (gmp_randstate_t state, gmp_randalg_t alg, ...)

Initialize random state variable state.
alg denotes what algorithm to use for random number generation.
Use one of
 GMP_RAND_ALG_LC  Linear congruential.
A fast generator defined by X = (aX + c) mod m.
A third argument size of type unsigned long int is required. size
is the size of the largest good quality random number to be generated,
expressed in number of bits. If the random generation functions are asked for
a bigger random number than indicated by this parameter, two or more numbers
of size bits will be generated and concatenated, resulting in a "bad"
random number. This can be used to generate big random numbers relatively
cheap if the quality of randomness isn't of great importance.
a, c, and m are picked from a table where the modulus (m) is a power of 2 and
the multiplier is congruent to 5 (mod 8). The choice is based on the
size parameter. The maximum size supported by this algorithm is
128. If you need bigger random numbers, use your own scheme and call one of
the other
gmp_randinit
functions.
If alg is 0 or GMP_RAND_ALG_DEFAULT, the default algorithm is used. The
default algorithm is typically a fast algorithm like the linear congruential
and requires a third size argument (see GMP_RAND_ALG_LC).
When you're done with a state variable, call gmp_randclear
to deallocate any memory allocated by this function.
gmp_randinit
may set the following bits in gmp_errno:
 GMP_ERROR_UNSUPPORTED_ARGUMENT  alg is unsupported
 GMP_ERROR_INVALID_ARGUMENT  size is too big
 Function: void gmp_randinit_lc_2exp (gmp_randstate_t state, mpz_t a,

unsigned long int c, unsigned long int m2exp)
Initialize random state variable state with given linear congruential
scheme.
Parameters a, c, and m2exp are the multiplier, adder, and
modulus for the linear congruential scheme to use, respectively. The modulus
is expressed as a power of 2, so that
@ifnottex
m = 2^m2exp.
The least significant bits of a random number generated by the linear
congruential algorithm where the modulus is a power of two are not very random.
Therefore, the lower half of a random number generated by an LC scheme
initialized with this function is discarded. Thus, the size of a random number
is m2exp / 2 (rounded upwards) bits when this function has been used for
initializing the random state.
When you're done with a state variable, call gmp_randclear
to deallocate any memory allocated by this function.
 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 the initial seed value.
Parameter seed is the initial random seed. The function
gmp_randseed_ui
takes the seed as an unsigned long int rather
than as an mpz_t.
 Function: void gmp_randclear (gmp_randstate_t state)

Free all memory occupied by state. Make sure to call this
function for all
gmp_randstate_t
variables when you are done with
them.
These functions are intended to be fully compatible with the Berkeley MP
library which is available on many BSD derived U*ix systems. The
`enablempbsd' option must be used when building GNU MP to make these
available (see section Installing GMP).
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 restrictioninputs 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, '\0'terminate
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 differentlyor not at allfor
negative arguments.
 Function: void msqrt (MINT *operand, MINT *root, MINT *remainder)

@ifnottex
Set root to the truncated integer part of the square root of
operand. Set remainder to
operandroot*root,
(i.e., zero if operand 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 *operand1, MINT *operand2, MINT *res)

Set res to the greatest common divisor of operand1 and
operand2.
 Function: int mcmp (MINT *operand1, MINT *operand2)

Compare operand1 and operand2. Return a positive value if
operand1 > operand2, zero if operand1 =
operand2, and a negative value if operand1 < operand2.
 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 *operand)

Convert operand 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.
 Function: void mfree (MINT *operand)

Deallocate, the space used by operand. This function should
only be passed a value returned by
itom
or xtom
.
By default, GMP uses malloc
, realloc
and free
for memory
allocation. If malloc
or realloc
fails, GMP prints a message to
the standard error output and terminates execution.
Some applications might want to allocate memory in other ways, or might not
want a fatal error when there is no more memory available. To accomplish
this, you can specify alternative memory allocation functions.
This can be done in the Berkeley compatibility library 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 retained.
Be sure to call this function only when there are no active GMP
objects allocated using the previous memory functions! Usually, that means
that you have to call this function before any other GMP function.
The functions you supply should fit the following declarations:
 Function: void * allocate_function (size_t alloc_size)

This function should return a pointer to newly allocated space with at least
alloc_size storage units.
 Function: void * reallocate_function (void *ptr, size_t old_size, size_t new_size)

This function should return a pointer to newly allocated space of at least
new_size storage units, after copying at least the first old_size
storage units from ptr. It should also deallocate the space at
ptr.
You can assume that the space at ptr was formerly returned from
allocate_function
or reallocate_function
, for a request for
old_size storage units.
 Function: void deallocate_function (void *ptr, size_t size)

Deallocate the space pointed to by ptr.
You can assume that the space at ptr was formerly returned from
allocate_function
or reallocate_function
, for a request for
size storage units.
(A storage unit is the unit in which the sizeof
operator returns
the size of an object, normally an 8 bit byte.)
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 3013141942.
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 functions mpz_jacobi
and mpz_legendre
.
Andreas Schwab contributed the files `mpn/m68k/lshift.S' and
`mpn/m68k/rshift.S'.
The development of floating point functions of GNU MP 2, were supported in part
by the ESPRITBRA (Basic Research Activities) 6846 project POSSO (POlynomial
System SOlving).
GNU MP 2 was finished and released by SWOX AB (formerly known as TMG
Datakonsult), Swedenborgsgatan 23, SE118 27 STOCKHOLM, 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 3way 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 BurnikelZiegler division code, the REDC code, the
REDCbased mpz_powm code, and the FFT multiply code. The ECMNET project Paul
is organizing has been a driving force behind many of the optimization of 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 wrote a lot of very high quality x86 code, optimized for most CPU
variants. He also made countless other valuable contributions.
Steve Root helped write the optimized alpha 21264 assembly code.
GNU MP 3.1 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 [email protected]
about the omission!)

Donald E. Knuth, "The Art of Computer Programming", vol 2,
"Seminumerical Algorithms", 3rd edition, AddisonWesley, 1988.

John D. Lipson, "Elements of Algebra and Algebraic Computing",
The Benjamin Cummings Publishing Company Inc, 1981.

Richard M. Stallman, "Using and Porting GCC", Free Software Foundation, 1999,
available online http://www.gnu.org/software/gcc/onlinedocs/, and in
the GCC package ftp://ftp.gnu.org/pub/gnu/gcc/.

Peter L. Montgomery, "Modular Multiplication Without Trial Division", in
Mathematics of Computation, volume 44, number 170, April 1985.

Torbjorn Granlund and Peter L. Montgomery, "Division by Invariant
Integers using Multiplication", in Proceedings of the SIGPLAN
PLDI'94 Conference, June 1994. Available online,
ftp://ftp.cwi.nl/pub/pmontgom/divcnst.psa4.gz (and .psl.gz too).

Tudor Jebelean,
"An algorithm for exact division",
Journal of Symbolic Computation,
v. 15, 1993, pp. 169180.
Research report version available online
ftp://ftp.risc.unilinz.ac.at/pub/techreports/1992/9235.ps.gz

Kenneth Weber, "The accelerated integer GCD algorithm",
ACM Transactions on Mathematical Software,
v. 21 (March), 1995, pp. 111122.

Christoph Burnikel and Joachim Ziegler, "Fast Recursive Division",
MaxPlanckInstitut fuer Informatik Research Report MPII981022,
http://www.mpisb.mpg.de/~ziegler/TechRep.ps.gz.

Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, "Handbook of
Applied Cryptography", http://cacr.math.uwaterloo.ca/hac/.

Henri Cohen, "A Course in Computational Algebraic Number Theory", Graduate
Texts in Mathematics number 138, SpringerVerlag, 1993. Errata available
online
http://www.math.ubordeaux.fr/~cohen
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ABI
About this manual
alloca
Allocation of memory
Anonymous FTP of latest version
Arithmetic functions, Arithmetic functions, Arithmetic functions
Assignment functions, Assignment functions
Basics
Berkeley MP compatible functions
Binomial coefficient functions
Bit manipulation functions
Bit shift left
Bit shift right
Bits per limb
BSD MP compatible functions
Bug reporting
Build notes for binary packaging
Build notes for particular systems
Build options
Build problems known
Comparison functions, Comparison functions, Comparison functions
Compatibility with older versions
Conditions for copying GNU MP
Configuring GMP
Constants
Contributors
Conventions for variables
Conversion functions, Conversion functions
Copying conditions
CPUs supported
Custom allocation
Demonstration programs
Division functions, Division functions, Division functions
Exact division functions
Example programs
Exponentiation functions, Exponentiation functions
Extended GCD
Factorial functions
Fibonacci sequence functions
Float arithmetic functions
Float assignment functions
Float comparison functions
Float conversion functions
Float functions
Float init and assign functions
Float initialization functions
Float input and output functions
Float miscellaneous functions
Floatingpoint functions
Floatingpoint number
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Function classes
GMP version number
`gmp.h'
Greatest common divisor functions
Home page
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Initialization and assignment functions, Initialization and assignment functions, Initialization and assignment functions
Initialization functions, Initialization functions
Input functions, Input functions, Input functions
Installing GMP
Integer
Integer arithmetic functions
Integer assignment functions
Integer bit manipulation functions
Integer comparison functions
Integer conversion functions
Integer division functions
Integer exponentiation functions
Integer functions
Integer init and assign
Integer initialization functions
Integer input and output functions
Integer miscellaneous functions
Integer random number functions
Integer root functions
Introduction
ISA
Jabobi symbol functions
Kronecker symbol functions
Latest version of GMP
Least common multiple functions
Libtool versioning
Limb
Limb size
Logical functions
Lowlevel functions
Mailing list
Memory allocation
Miscellaneous float functions
Miscellaneous integer functions
Miscellaneous rational functions
Modular inverse functions
`mp.h'
Multithreading
Nomenclature
Number theoretic functions
Numerator and denominator
Output functions, Output functions, Output functions
Packaged builds
Parameter conventions
Precision of floats
Prime testing functions
Random number functions, Random number functions
Random number state
Rational arithmetic functions
Rational comparison functions
Rational init and assign
Rational input and output functions
Rational miscellaneous functions
Rational number
Rational number functions
Rational numerator and denominator
Reentrancy
References
Reporting bugs
Root extraction functions, Root extraction functions
Stack overflow segfaults
Stripped libraries
Thread safety
Types
Upward compatibility
Useful macros and constants
Userdefined precision
Variable conventions
Version number
Web page
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__GNU_MP_VERSION
__GNU_MP_VERSION_MINOR
__GNU_MP_VERSION_PATCHLEVEL
_mpz_realloc
allocate_function
deallocate_function
gcd
gmp_randclear
gmp_randinit
gmp_randinit_lc_2exp
gmp_randseed
gmp_randseed_ui
itom
madd
mcmp
mdiv
mfree
min
mout
move
mp_limb_t
mp_set_memory_functions
mpf_abs
mpf_add
mpf_add_ui
mpf_ceil
mpf_clear
mpf_cmp
mpf_cmp_si
mpf_cmp_ui
mpf_div
mpf_div_2exp
mpf_div_ui
mpf_eq
mpf_floor
mpf_get_d
mpf_get_prec
mpf_get_str
mpf_init
mpf_init2
mpf_init_set
mpf_init_set_d
mpf_init_set_si
mpf_init_set_str
mpf_init_set_ui
mpf_inp_str
mpf_mul
mpf_mul_2exp
mpf_mul_ui
mpf_neg
mpf_out_str
mpf_pow_ui
mpf_random2
mpf_reldiff
mpf_set
mpf_set_d
mpf_set_default_prec
mpf_set_prec
mpf_set_prec_raw
mpf_set_q
mpf_set_si
mpf_set_str
mpf_set_ui
mpf_set_z
mpf_sgn
mpf_sqrt
mpf_sqrt_ui
mpf_sub
mpf_sub_ui
mpf_swap
mpf_t
mpf_trunc
mpf_ui_div
mpf_ui_sub
mpf_urandomb
mpn_add
mpn_add_1
mpn_add_n
mpn_addmul_1
mpn_bdivmod
mpn_cmp
mpn_divexact_by3
mpn_divexact_by3c
mpn_divmod
mpn_divmod_1
mpn_divrem
mpn_divrem_1
mpn_gcd
mpn_gcd_1
mpn_gcdext
mpn_get_str
mpn_hamdist
mpn_lshift
mpn_mod_1
mpn_mul
mpn_mul_1
mpn_mul_n
mpn_perfect_square_p
mpn_popcount
mpn_preinv_mod_1
mpn_random
mpn_random2
mpn_rshift
mpn_scan0
mpn_scan1
mpn_set_str
mpn_sqrtrem
mpn_sub
mpn_sub_1
mpn_sub_n
mpn_submul_1
mpn_tdiv_qr
mpq_add
mpq_canonicalize
mpq_clear
mpq_cmp
mpq_cmp_ui
mpq_denref
mpq_div
mpq_equal
mpq_get_d
mpq_get_den
mpq_get_num
mpq_init
mpq_inv
mpq_mul
mpq_neg
mpq_numref
mpq_out_str
mpq_set
mpq_set_d
mpq_set_den
mpq_set_num
mpq_set_si
mpq_set_ui
mpq_set_z
mpq_sgn
mpq_sub
mpq_swap
mpq_t
mpz_abs
mpz_add
mpz_add_ui
mpz_addmul_ui
mpz_and
mpz_array_init
mpz_bin_ui
mpz_bin_uiui
mpz_cdiv_q
mpz_cdiv_q_ui
mpz_cdiv_qr
mpz_cdiv_qr_ui
mpz_cdiv_r
mpz_cdiv_r_ui
mpz_cdiv_ui
mpz_clear
mpz_clrbit
mpz_cmp
mpz_cmp_si
mpz_cmp_ui
mpz_cmpabs
mpz_cmpabs_ui
mpz_com
mpz_divexact
mpz_even_p
mpz_fac_ui
mpz_fdiv_q
mpz_fdiv_q_2exp
mpz_fdiv_q_ui
mpz_fdiv_qr
mpz_fdiv_qr_ui
mpz_fdiv_r
mpz_fdiv_r_2exp
mpz_fdiv_r_ui
mpz_fdiv_ui
mpz_fib_ui
mpz_fits_sint_p
mpz_fits_slong_p
mpz_fits_sshort_p
mpz_fits_uint_p
mpz_fits_ulong_p
mpz_fits_ushort_p
mpz_gcd
mpz_gcd_ui
mpz_gcdext
mpz_get_d
mpz_get_si
mpz_get_str
mpz_get_ui
mpz_getlimbn
mpz_hamdist
mpz_init
mpz_init_set
mpz_init_set_d
mpz_init_set_si
mpz_init_set_str
mpz_init_set_ui
mpz_inp_raw
mpz_inp_str
mpz_invert
mpz_ior
mpz_jacobi
mpz_kronecker_si
mpz_kronecker_ui
mpz_lcm
mpz_legendre
mpz_mod
mpz_mod_ui
mpz_mul
mpz_mul_2exp
mpz_mul_si
mpz_mul_ui
mpz_neg
mpz_nextprime
mpz_odd_p
mpz_out_raw
mpz_out_str
mpz_perfect_power_p
mpz_perfect_square_p
mpz_popcount
mpz_pow_ui
mpz_powm
mpz_powm_ui
mpz_probab_prime_p
mpz_random
mpz_random2
mpz_remove
mpz_root
mpz_rrandomb
mpz_scan0
mpz_scan1
mpz_set
mpz_set_d
mpz_set_f
mpz_set_q
mpz_set_si
mpz_set_str
mpz_set_ui
mpz_setbit
mpz_sgn
mpz_si_kronecker
mpz_size
mpz_sizeinbase
mpz_sqrt
mpz_sqrtrem
mpz_sub
mpz_sub_ui
mpz_swap
mpz_t
mpz_tdiv_q
mpz_tdiv_q_2exp
mpz_tdiv_q_ui
mpz_tdiv_qr
mpz_tdiv_qr_ui
mpz_tdiv_r
mpz_tdiv_r_2exp
mpz_tdiv_r_ui
mpz_tdiv_ui
mpz_tstbit
mpz_ui_kronecker
mpz_ui_pow_ui
mpz_urandomb
mpz_urandomm
mpz_xor
msqrt
msub
mtox
mult
pow
reallocate_function
rpow
sdiv
xtom
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