Dana Jacobsen >
Math-Prime-Util-GMP-0.11 >
Math::Prime::Util::GMP

Module Version: 0.11
- NAME
- VERSION
- SYNOPSIS
- DESCRIPTION
- FUNCTIONS
- is_prob_prime
- is_prime
- is_provable_prime
- is_provable_prime_with_cert
- is_strong_pseudoprime
- is_strong_lucas_pseudoprime
- is_aks_prime
- is_nminus1_prime
- is_ecpp_prime
- primes
- next_prime
- prev_prime
- primorial
- pn_primorial
- consecutive_integer_lcm
- factor
- trial_factor
- prho_factor
- pbrent_factor
- pminus1_factor
- holf_factor
- squfof_factor
- ecm_factor
- qs_factor

- SEE ALSO
- REFERENCES
- AUTHORS
- ACKNOWLEDGEMENTS
- COPYRIGHT

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

Version 0.11

use Math::Prime::Util::GMP ':all'; my $n = "115792089237316195423570985008687907853269984665640564039457584007913129639937"; # This doesn't impact the operation of the module at all, but does let you # enter big number arguments directly as well as enter (e.g.): 2**2048 + 1. use bigint; # These return 0 for composite, 2 for prime, and 1 for probably prime # Numbers under 2^64 will return 0 or 2. # is_prob_prime does a BPSW primality test for numbers > 2^64 # is_prime adds some MR tests and a quick test to try to prove the result # is_provable_prime will spend a lot of effort on proving primality say "$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($n); # o1 = B1, o2 = # of curves @factors = qs_factor($n); # (no arguments)

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.

As with "is_prob_prime", a BPSW test is first performed. If this indicates "probably prime" then a small number of Miller-Rabin tests with random bases are performed. For numbers under 200 bits, a quick BLS75 `n-1`

primality proof is attempted. This is tuned to give up if the result cannot be quickly determined, and results in approximately 30% success rate at 128-bits.

The result is that many numbers will return back 2 (definitely prime), and the numbers that return 1 (probably prime) have gone through more tests than "is_prob_prime" while not taking too long.

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 first uses BPSW and a small number of Miller-Rabin tests with random bases to weed out composites and provide a deterministic answer for tiny numbers (under `2^64`

). A quick BLS75 `n-1`

test is attempted, followed by ECPP.

The time required for primes of different input sizes on a circa-2009 workstation averages about `3ms`

for 30-digits, `8ms`

for 40-digit, `30ms`

for 60-digit, `75ms`

for 80-digit, `200ms`

for 100-digit, `2s`

for 200-digit, and 400-digit inputs about a minute. Expect a lot of time variation for larger inputs. You can see progress indication if verbose is turned on (some at level 1, and a lot at level 2).

A certificate can be obtained along with the result using the "is_provable_prime_with_cert" method.

Takes a positive number as input and returns back an array with two elements. The result will be one of:

(0, []) The input is composite. (1, []) The input is probably prime but we could not prove it. This is a failure in our ability to factor some necessary element in a reasonable time, not a significant proof failure. (2, [cert] ) The input is prime, and the certificate contains all the information necessary to verify this.

The certificate can be run through the "verify_prime" in Math::Prime::Util function of Math::Prime::Util for verification. The current implementation will return either certificates of either "n-1" type, "ECPP" type, or a mix of the two (e.g. ECPP ended with n-1 for the final number).

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. The base must be a positive integer.

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. If base modulo n <= 1 or base modulo n = n-1, then the result will be 1. This allows the bases to be larger than n if desired, while still returning meaningful results. For example,

is_strong_pseudoprime(367, 1101)

would incorrectly return 0 if this was not done properly. A 0 result should be returned only if n is composite, regardless of the base.

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, "is_nminus1_prime" and "is_ecpp_prime" are **much** faster.

Typically you should use "is_provable_prime" and let it decide the method.

say "$n is definitely prime" if is_nminus1_prime($n);

Takes a positive number as input, and returns 1 if the input passes either theorem 5 or theorem 7 of the Brillhart-Lehmer-Selfridge primality test. This is a deterministic unconditional primality test which requires factoring `n-1`

to a linear factor less than the cube root of the input. For small inputs (under 40 digits) this is typically very easy, and some numbers will naturally lead to this being very fast. As the input grows, this method slows down rapidly.

Typically you should use "is_provable_prime" and let it decide the method.

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

Takes a positive number as input, and returns 1 if the input passes the ECPP primality test. This is the Atkin-Morain Elliptic Curve Primality Proving algorithm. It is the fastest primality proving method in Math::Prime::Util.

This implementation uses a "factor all strategy" (FAS) with backtracking. A limited set of about 500 precalculated discriminants are used, which works well for inputs up to 300 digits, and for many inputs up to one thousand digits. Having a larger set will help with large numbers (a set of 2650 is available on github in the `xt/`

directory). A future implementation may include code to generate class polynomials as needed.

Typically you should use "is_provable_prime" and let it decide the method.

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 (a Fermat variant), ECM (elliptic curve method), and QS (quadratic sieve). Certainly improvements could be designed for this algorithm (suggestions are welcome).

In practice, this factors 26-digit semiprimes in under `100ms`

, 36-digit semiprimes in under one second. Arbitrary integers are factored faster. It is many orders of magnitude faster than any other factoring module on CPAN circa early 2013. It is comparable in speed to Math::Pari's `factorint`

for most inputs.

If you want better factoring in general, I recommend looking at the standalone programs yafu, msieve, gmp-ecm, and 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); # B1 = 12500 my @factors = ecm_factor($n, 12500, 10); # B1 = 12500, curves = 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 an implementation of Hendrik Lenstra's elliptic curve factoring method, usually referred to as ECM. The implementation is reasonable, using projective coordinates, Montgomery's PRAC heuristic for EC multiplication, and two stages. It is much slower than the latest GMP-ECM, but still quite useful for factoring reasonably sized inputs.

my @factors = qs_factor($n);

Given a positive number input, tries to discover factors using QS (the quadratic sieve). The resulting array will contain one or more numbers such that multiplying @factors yields the original input. Typically multiple factors will be produced, unlike the other `..._factor`

routines.

The current implementation is a modified version of SIMPQS, a predecessor to the QS in FLINT, and was written by William Hart in 2006. It will not operate on input less than 30 digits. The memory use for large inputs is more than desired, so other methods such as "pbrent_factor", "pminus1_factor", and "ecm_factor" are recommended to begin with to filter out small factors. However, it is substantially faster than the other methods on large inputs having large factors, and is the method of choice for 35+ digit semiprimes.

- Math::Prime::Util. Has many more functions, lots of fast code for dealing with native-precision arguments (including much faster primes using sieves), and will use this module when needed for big numbers. Using Math::Prime::Util rather than this module directly is recommended.
- Math::Primality (version 0.08) 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.
- Math::Pari Supports quite a bit of the same functionality (and much more). See "SEE ALSO" in Math::Prime::Util for more detailed information on how the modules compare.
- yafu, msieve, gmp-ecm, GGNFS Good general purpose factoring utilities. These will be faster than this module, and
**much**better as the factor increases in size. - Primo is the state of the art in freely available (though not open source!) primality proving programs. If you have 1000+ digit numbers to prove, you want to use this.

- 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
- A.O.L. Atkin and F. Morain, "Elliptic Curves and primality proving", Mathematics of Computation, v61, 1993, pages 29-68. http://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1199989-X/
- R.G.E. Pinch, "Some Primality Testing Algorithms", June 1993. Describes the primality testing methods used by many CAS systems and how most were compromised. Gives recommendations for primality testing APIs.

Dana Jacobsen <dana@acm.org>

William Hart wrote the SIMPQS code which is the basis for the QS code.

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., Atkin, 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 Buhrow 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.

Paul Zimmermann's papers and GMP-ECM code were of great value for my projective ECM implementation, as well as the papers by Brent and Montgomery.

Copyright 2011-2013 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.

SIMPQS Copyright 2006, William Hart. SIMPQS is distributed under GPL v2+.

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