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Math::Prime::Util::GMP

Module Version: 0.06
Math::Prime::Util::GMP - Utilities related to prime numbers and factoring, using GMP

Version 0.06

use Math::Prime::Util::GMP ':all'; my $n = "115792089237316195423570985008687907853269984665640564039457584007913129639937"; # This doesn't impact the operation of the module at all, but does let you # enter big number arguments directly as well as enter (e.g.): 2**2048 + 1. use bigint; # These return 0 for composite, 2 for prime, and 1 for probably prime # Numbers under 2^64 will return 0 or 2. # is_prob_prime does a BPSW primality test for numbers > 2^64 # is_prime adds a quick test to try to prove the result # is_provable_prime will spend a lot of effort on proving primality say "$n is probably prime" if is_prob_prime($n); say "$n is ", qw(composite prob_prime def_prime)[is_prime($n)]; say "$n is definitely prime" if is_provable_prime($n) == 2; # Miller-Rabin and strong Lucas-Selfridge pseudoprime tests say "$n is a prime or spsp-2/7/61" if is_strong_pseudoprime($n, 2, 7, 61); say "$n is a prime or slpsp" if is_strong_lucas_pseudoprime($n); # Return array reference to primes in a range. my $aref = primes( 10 ** 200, 10 ** 200 + 10000 ); $next = next_prime($n); # next prime > n $prev = prev_prime($n); # previous prime < n # Primorials and lcm say "23# is ", primorial(23); say "The product of the first 47 primes is ", pn_primorial(47); say "lcm(1..1000) is ", consecutive_integer_lcm(1000); # Find prime factors of big numbers @factors = factor(5465610891074107968111136514192945634873647594456118359804135903459867604844945580205745718497); # Finer control over factoring. # These stop after finding one factor or exceeding their limit. # # optional arguments o1, o2, ... @factors = trial_factor($n); # test up to o1 @factors = prho_factor($n); # no more than o1 rounds @factors = pbrent_factor($n); # no more than o1 rounds @factors = holf_factor($n); # no more than o1 rounds @factors = squfof_factor($n); # no more than o1 rounds @factors = pminus1_factor($n); # o1 = smoothness limit, o2 = stage 2 limit @factors = ecm_factor($n); # o1 = B1, o2 = # of curves

A set of utilities related to prime numbers, using GMP. This includes primality tests, getting primes in a range, and factoring.

While it certainly can be used directly, the main purpose of this module is for Math::Prime::Util. That module will automatically load this if it is installed, greatly speeding up many of its operations on big numbers.

Inputs and outputs for big numbers are via strings, so you do not need to use a bigint package in your program. However if you do use bigints, inputs will be converted internally so there is no need to convert before a call. Output results are returned as either Perl scalars (for native-size) or strings (for bigints). Math::Prime::Util tries to reconvert all strings back into the callers bigint type if possible, which makes it more convenient for calculations.

my $prob_prime = is_prob_prime($n); # Returns 0 (composite), 2 (prime), or 1 (probably prime)

Takes a positive number as input and returns back either 0 (composite), 2 (definitely prime), or 1 (probably prime).

For inputs below `2^64`

a deterministic test is performed, so the possible return values are 0 (composite) or 2 (definitely prime).

For inputs above `2^64`

, a probabilistic test is performed. Only 0 (composite) and 1 (probably prime) are returned. The current implementation uses a strong Baillie-PSW test. There is a possibility that composites may be returned marked prime, but since the test was published in 1980, not a single BPSW pseudoprime has been found, so it is extremely likely to be prime. While we believe (Pomerance 1984) that an infinite number of counterexamples exist, there is a weak conjecture (Martin) that none exist under 10000 digits.

say "$n is prime!" if is_prime($n);

Takes a positive number as input and returns back either 0 (composite), 2 (definitely prime), or 1 (probably prime). Composites will act exactly like `is_prob_prime`

, as will numbers less than `2^64`

. For numbers larger than `2^64`

, some additional tests are performed on probable primes to see if they can be proven by another means.

Currently the the method used once numbers have been marked probably prime by BPSW is the BLS75 method: Brillhart, Lehmer, and Selfridge's improvement to the Pocklington-Lehmer primality test. The test requires factoring `n-1`

to `(n/2)^(1/3)`

, compared to `n^(1/2)`

of the standard Pocklington-Lehmer or PPBLS test, or a complete factoring for the Lucas test. The main problem is still finding factors, which is done using a small number of rounds of Pollard's Rho. This works quite well and is very fast when the factors are small.

say "$n is definitely prime!" if is_provable_prime($n) == 2;

Takes a positive number as input and returns back either 0 (composite), 2 (definitely prime), or 1 (probably prime). A great deal of effort is taken to return either 0 or 2 for all numbers.

The current method is the BLS75 algorithm as described in `is_prime`

, but using much more aggressive factoring. Planned enhancements for a later release include using a faster method (e.g. APRCL or ECPP), and the ability to return a certificate.

my $maybe_prime = is_strong_pseudoprime($n, 2); my $probably_prime = is_strong_pseudoprime($n, 2, 3, 5, 7, 11, 13, 17);

Takes a positive number as input and one or more bases. Returns 1 if the input is a prime or a strong pseudoprime to all of the bases, and 0 if not.

If 0 is returned, then the number really is a composite. If 1 is returned, then it is either a prime or a strong pseudoprime to all the given bases. Given enough distinct bases, the chances become very strong that the number number is actually prime.

Both the input number and the bases may be big integers. The bases must be greater than 1, however they may be as large as desired.

This is usually used in combination with other tests to make either stronger tests (e.g. the strong BPSW test) or deterministic results for numbers less than some verified limit (e.g. Jaeschke showed in 1993 that no more than three selected bases are required to give correct primality test results for any 32-bit number). Given the small chances of passing multiple bases, there are some math packages that just use multiple MR tests for primality testing.

Even numbers other than 2 will always return 0 (composite). While the algorithm works with even input, most sources define it only on odd input. Returning composite for all non-2 even input makes the function match most other implementations including Math::Primality's `is_strong_pseudoprime`

function.

Takes a positive number as input, and returns 1 if the input is a strong Lucas pseudoprime using the Selfridge method of choosing D, P, and Q (some sources call this a strong Lucas-Selfridge pseudoprime). This is one half of the BPSW primality test (the Miller-Rabin strong pseudoprime test with base 2 being the other half).

say "$n is definitely prime" if is_aks_prime($n);

Takes a positive number as input, and returns 1 if the input passes the Agrawal-Kayal-Saxena (AKS) primality test. This is a deterministic unconditional primality test which runs in polynomial time for general input. In practice, the BLS75 method used by is_provable_prime is much faster.

my $aref1 = primes( 1_000_000 ); my $aref2 = primes( 2 ** 448, 2 ** 448 + 10000 ); say join ",", @{primes( 2**2048, 2**2048 + 10000 )};

Returns all the primes between the lower and upper limits (inclusive), with a lower limit of `2`

if none is given.

An array reference is returned (with large lists this is much faster and uses less memory than returning an array directly).

The current implementation uses repeated calls to `next_prime`

, which is good for very small ranges, but not good for large ranges. A future release may use a multi-segmented sieve when appropriate.

$n = next_prime($n);

Returns the next prime greater than the input number.

$n = prev_prime($n);

Returns the prime smaller than the input number. 0 is returned if the input is `2`

or lower.

$p = primorial($n);

Given an unsigned integer argument, returns the product of the prime numbers which are less than or equal to `n`

. This definition of `n#`

follows OEIS series A034386 and Wikipedia: Primorial definition for natural numbers.

$p = pn_primorial($n)

Given an unsigned integer argument, returns the product of the first `n`

prime numbers. This definition of `p_n#`

follows OEIS series A002110 and Wikipedia: Primorial definition for prime numbers.

The two are related with the relationships:

pn_primorial($n) == primorial( nth_prime($n) ) primorial($n) == pn_primorial( prime_count($n) )

$lcm = consecutive_integer_lcm($n);

Given an unsigned integer argument, returns the least common multiple of all integers from 1 to `n`

. This can be done by manipulation of the primes up to `n`

, resulting in much faster and memory-friendly results than using n factorial.

@factors = factor(640552686568398413516426919223357728279912327120302109778516984973296910867431808451611740398561987580967216226094312377767778241368426651540749005659); # Returns an array of 11 factors

Returns a list of prime factors of a positive number, in numerical order. The special cases of `n = 0`

and `n = 1`

will return `n`

.

Like most advanced factoring programs, a mix of methods is used. This includes trial division for small factors, perfect power detection, Pollard's Rho, Pollard's P-1 with various smoothness and stage settings, Hart's OLF, and ECM (elliptic curve method).

Certainly improvements could be designed for this algorithm (suggestions are welcome). Most importantly, improving ECM and adding MPQS/SIQS would be a big help with larger numbers. These are non-trivial (though feasible) methods.

In practice, this factors most 26-digit semiprimes in under a second. It is many orders of magnitude faster than any other factoring module on CPAN circa 2012. Pari's factorint is faster (and can be accessed from Perl via Math::Pari), as are the standalone programs yafu, msieve, gmp-ecm, GGNFS.

my @factors = trial_factor($n); my @factors = trial_factor($n, 1000);

Given a positive number input, tries to discover a factor using trial division. The resulting array will contain either two factors (it succeeded) or the original number (no factor was found). In either case, multiplying @factors yields the original input. An optional divisor limit may be given as the second parameter. Factoring will stop when the input is a prime, one factor is found, or the input has been tested for divisibility with all primes less than or equal to the limit. If no limit is given, then `2**31-1`

will be used.

This is a good and fast initial test, and will be very fast for small numbers (e.g. under 1 million). It becomes unreasonably slow in the general case as the input size increases.

my @factors = prho_factor($n); my @factors = prho_factor($n, 100_000_000);

Given a positive number input, tries to discover a factor using Pollard's Rho method. The resulting array will contain either two factors (it succeeded) or the original number (no factor was found). In either case, multiplying @factors yields the original input. An optional number of rounds may be given as the second parameter. Factoring will stop when the input is a prime, one factor has been found, or the number of rounds has been exceeded.

This is the Pollard Rho method with `f = x^2 + 3`

and default rounds 64M. It is very good at finding small factors.

my @factors = pbrent_factor($n); my @factors = pbrent_factor($n, 100_000_000);

Given a positive number input, tries to discover a factor using Pollard's Rho method with Brent's algorithm. The resulting array will contain either two factors (it succeeded) or the original number (no factor was found). In either case, multiplying @factors yields the original input. An optional number of rounds may be given as the second parameter. Factoring will stop when the input is a prime, one factor has been found, or the number of rounds has been exceeded.

This is the Pollard Rho method using Brent's modified cycle detection and backtracking. It is essentially Algorithm P''2 from Brent (1980). Parameters used are `f = x^2 + 3`

and default rounds 64M. It is very good at finding small factors.

my @factors = pminus1_factor($n); # Set B1 smoothness to 10M, second stage automatically set. my @factors = pminus1_factor($n, 10_000_000); # Run p-1 with B1 = 10M, B2 = 100M. my @factors = pminus1_factor($n, 10_000_000, 100_000_000);

Given a positive number input, tries to discover a factor using Pollard's `p-1`

method. The resulting array will contain either two factors (it succeeded) or the original number (no factor was found). In either case, multiplying @factors yields the original input. An optional first stage smoothness factor (B1) may be given as the second parameter. This will be the smoothness limit B1 for for the first stage, and will use `10*B1`

for the second stage limit B2. If a third parameter is given, it will be used as the second stage limit B2. Factoring will stop when the input is a prime, one factor has been found, or the algorithm fails to find a factor with the given smoothness.

This is Pollard's `p-1`

method using a default smoothness of 5M and a second stage of `B2 = 10 * B1`

. It can quickly find a factor `p`

of the input `n`

if the number `p-1`

factors into small primes. For example `n = 22095311209999409685885162322219`

has the factor `p = 3916587618943361`

, where `p-1 = 2^7 * 5 * 47 * 59 * 3137 * 703499`

, so this method will find a factor in the first stage if `B1 >= 703499`

or in the second stage if `B1 >= 3137`

and `B2 >= 703499`

.

The implementation is written from scratch using the basic algorithm including a second stage as described in Montgomery 1987. It is faster than most simple implementations I have seen (many of which are written assuming native precision inputs), but slower than Ben Buhrow's code used in earlier versions of yafu, and nowhere close to the speed of the version included with modern GMP-ECM.

my @factors = holf_factor($n); my @factors = holf_factor($n, 100_000_000);

Given a positive number input, tries to discover a factor using Hart's OLF method. The resulting array will contain either two factors (it succeeded) or the original number (no factor was found). In either case, multiplying @factors yields the original input. An optional number of rounds may be given as the second parameter. Factoring will stop when the input is a prime, one factor has been found, or the number of rounds has been exceeded.

This is Hart's One Line Factorization method, which is a variant of Fermat's algorithm. A premultiplier of 480 is used. It is very good at factoring numbers that are close to perfect squares, or small numbers. Very naive methods of picking RSA parameters sometimes yield numbers in this form, so it can be useful to run this a few rounds to check. For example, the number:

18548676741817250104151622545580576823736636896432849057 \ 10984160646722888555430591384041316374473729421512365598 \ 29709849969346650897776687202384767704706338162219624578 \ 777915220190863619885201763980069247978050169295918863

was proposed by someone as an RSA key. It is indeed composed of two distinct prime numbers of similar bit length. Most factoring methods will take a **very** long time to break this. However one factor is almost exactly 5x larger than the other, allowing HOLF to factor this 222-digit semiprime in only a few milliseconds.

my @factors = squfof_factor($n); my @factors = squfof_factor($n, 100_000_000);

Given a positive number input, tries to discover a factor using Shanks' square forms factorization method (usually known as SQUFOF). The resulting array will contain either two factors (it succeeded) or the original number (no factor was found). In either case, multiplying @factors yields the original input. An optional number of rounds may be given as the second parameter. Factoring will stop when the input is a prime, one factor has been found, or the number of rounds has been exceeded.

This is Daniel Shanks' SQUFOF (square forms factorization) algorithm. The particular implementation is a non-racing multiple-multiplier version, based on code ideas of Ben Buhrow and Jason Papadopoulos as well as many others. SQUFOF is often the preferred method for small numbers, and Math::Prime::Util as well as many other packages use it was the default method for native size (e.g. 32-bit or 64-bit) numbers after trial division. The GMP version used in this module will work for larger values, but my testing is showing that it is not faster than the `prho`

and `pbrent`

methods in general.

my @factors = ecm_factor($n); my @factors = ecm_factor($n, 12500); my @factors = ecm_factor($n, 12500, 10);

Given a positive number input, tries to discover a factor using ECM. The resulting array will contain either two factors (it succeeded) or the original number (no factor was found). In either case, multiplying @factors yields the original input. An optional maximum smoothness may be given as the second parameter, which relates to the size of factor to search for. An optional third parameter indicates the number of random curves to use at each smoothness value being searched.

This is a straightforward implementation of Hendrik Lenstra's elliptic curve factoring method, usually referred to as ECM. Its implementation is textbook, with no substantial optimizations done. It uses a single stage, affine coordinates, binary ladder multiplication, and simple initialization. The list of enhancements that can be made is numerous, and it will be much, much slower than GMP-ECM. However, it uses simple GMP and extends the useful factoring range of this module.

- Math::Prime::Util. Has many more functions, lots of good code for dealing with native-precision arguments (including much faster primes using sieves), and will use this module behind the scenes when needed for big numbers.
- Math::Primality (version 0.07) A Perl module with support for the strong Miller-Rabin test, strong Lucas-Selfridge test, the BPSW probable prime test, next_prime / prev_prime, the AKS primality test, and prime_count. It uses Math::GMPz to do all the calculations, so is faster than pure Perl bignums, but a little slower than XS+GMP. The prime_count function is only usable for very small inputs, but the other functions are quite good for big numbers. Make sure to use version 0.05 or newer.
- yafu, msieve, gmp-ecm, GGNFS Good general purpose factoring utilities. These will be faster than this module, and
**much**faster as the factor increases in size.

- Robert Baillie and Samuel S. Wagstaff, Jr., "Lucas Pseudoprimes", Mathematics of Computation, v35 n152, October 1980, pp 1391-1417. http://mpqs.free.fr/LucasPseudoprimes.pdf
- John Brillhart, D. H. Lehmer, and J. L. Selfridge, "New Primality Criteria and Factorizations of 2^m +/- 1", Mathematics of Computation, v29, n130, Apr 1975, pp 620-647. http://www.ams.org/journals/mcom/1975-29-130/S0025-5718-1975-0384673-1/S0025-5718-1975-0384673-1.pdf
- Richard P. Brent, "An improved Monte Carlo factorization algorithm", BIT 20, 1980, pp. 176-184. http://www.cs.ox.ac.uk/people/richard.brent/pd/rpb051i.pdf
- Peter L. Montgomery, "Speeding the Pollard and Elliptic Curve Methods of Factorization", Mathematics of Computation, v48, n177, Jan 1987, pp 243-264. http://www.ams.org/journals/mcom/1987-48-177/S0025-5718-1987-0866113-7/
- Richard P. Brent, "Parallel Algorithms for Integer Factorisation", in Number Theory and Cryptography, Cambridge University Press, 1990, pp 26-37. http://www.cs.ox.ac.uk/people/richard.brent/pd/rpb115.pdf
- Richard P. Brent, "Some Parallel Algorithms for Integer Factorisation", in Proc. Third Australian Supercomputer Conference, 1999. (Note: there are multiple versions of this paper) http://www.cs.ox.ac.uk/people/richard.brent/pd/rpb193.pdf
- William B. Hart, "A One Line Factoring Algorithm", preprint. http://wstein.org/home/wstein/www/home/wbhart/onelinefactor.pdf
- Daniel Shanks, "SQUFOF notes", unpublished notes, transcribed by Stephen McMath. http://www.usna.edu/Users/math/wdj/mcmath/shanks_squfof.pdf
- Jason E. Gower and Samuel S. Wagstaff, Jr, "Square Form Factorization", Mathematics of Computation, v77, 2008, pages 551-588. http://homes.cerias.purdue.edu/~ssw/squfof.pdf

Dana Jacobsen <dana@acm.org>

Obviously none of this would be possible without the mathematicians who created and published their work. Eratosthenes, Gauss, Euler, Riemann, Fermat, Lucas, Baillie, Pollard, Brent, Montgomery, Shanks, Hart, Wagstaff, Dixon, Pomerance, A.K Lenstra, H. W. Lenstra Jr., Knuth, etc.

The GNU GMP team, whose product allows me to concentrate on coding high-level algorithms and not worry about any of the details of how modular exponentiation and the like happen, and still get decent performance for my purposes.

Ben Buhrows and Jason Papadopoulos deserve special mention for their open source factoring tools, which are both readable and fast. In particular I am leveraging their SQUFOF work in the current implementation. They are a huge resource to the community.

Jonathan Leto and Bob Kuo, who wrote and distributed the Math::Primality module on CPAN. Their implementation of BPSW provided the motivation I needed to get it done in this module and Math::Prime::Util. I also used their module quite a bit for testing against.

Copyright 2011-2012 by Dana Jacobsen <dana@acm.org>

This program is free software; you can redistribute it and/or modify it under the same terms as Perl itself.

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