man Math::Prime::Util::GMP (3): Utilities related to prime numbers and factoring, using GMP

VERSION

Version 0.34

SYNOPSIS

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($n);
say "$n is a prime or eslpsp" if is_extra_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)

DESCRIPTION

A module for number theory in Perl using GMP. This includes primality tests, getting primes in a range, factoring, and more.

While it certainly can be used directly, the main purpose of this module is for Math::Prime::Util. That module will automatically load this one 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.

The various "is_*_pseudoprime" tests are more appropriately called "is_*_probable_prime" or "is_*_prp". They return 1 if the input is a probable prime based on their test. The naming convention is historical and follows Pari, Math::Primality, and some other math packages. The modern definition of pseudoprime is a composite that passes the test, rather than any number.

FUNCTIONS

is_prob_prime

  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" the test is deterministic, 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 the Baillie-PSW (BPSW) 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.

In more detail, we are using the extra-strong Lucas test (Grantham 2000) using the Baillie parameter selection method (see OEIS A217719). Previous versions of this module used the strong Lucas test with Selfridge parameters, but the extra-strong version produces fewer pseudoprimes while running 1.2 - 1.5x faster. It is slightly stronger than the test used in Pari <http://pari.math.u-bordeaux.fr/faq.html#primetest>.

is_prime

  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.

This call walks the line between the performance of ``is_prob_prime'' and the certainty of ``is_provable_prime''. Those calls may be more appropriate in some cases. What this function does is give most of the performance of the former, while adding more certainty. For finer tuning of this tradeoff, especially if performance is critical for 65- to 200-bit inputs, you may instead use ``is_prob_prime'' followed by additional probable prime tests such as ``miller_rabin_random'' and/or ``is_frobenius_underwood_pseudoprime''.

As with ``is_prob_prime'', a BPSW test is first performed. This is deterministic for all 64-bit numbers. Next, if the number is a Proth or LLR form, then a proof is constructed. If the result is still ``probably prime'' and the input is smaller than the Sorenson/Webster (2015) deterministic Miller-Rabin limit (approximately 82 bits) then the 11 or 12 Miller-Rabin tests are performed and the result is confirmed. For larger inputs that are still ``probably prime'' but 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 success rates of ~80% at 80 bits, ~30% at 128 bits, and ~13% at 160 bits. Lastly, for results still ``probably prime'', a small number of Miller-Rabin tests with random bases are performed.

The result is that many numbers will return 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.

For cryptographic key generation, you may want even more testing for probable primes (NIST recommends a few more additional M-R tests than we perform). The function ``miller_rabin_random'' is made for this. Alternately, a different test such as ``is_frobenius_underwood_pseudoprime'' can be used. Even better, use ``is_provable_prime'' which should be reasonably fast for sizes under 2048 bits. Typically for key generation one wants random primes, and there are many functions for that.

is_provable_prime

  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 to find composites and provide a deterministic answer for tiny numbers (under "2^64"). If no certificate is required, LLR and Proth tests can be run, and small numbers (under approximately "2^82") can be satisfied with a deterministic Miller-Rabin test. If the result is still not determined, 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, "5ms" for 40-digit, "20ms" for 60-digit, "50ms" for 80-digit, "100ms" 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. There is no appreciable extra performance cost for returning a certificate.

is_provable_prime_with_cert

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 (in other words, it remains a probable prime).
  (2, '...')   The input is prime, and the certificate contains all the
               information necessary to verify this.

The certificate is a text representation containing all the necessary information to verify the primality of the input in a reasonable time. The result can be used with ``verify_prime'' in Math::Prime::Util for verification. Proof types used include:

  ECPP
  BLS3
  BLS15
  BLS5
  Small

is_pseudoprime

Takes a positive number "n" and a base "a" as input, and returns 1 if "n" is a probable prime to base "a". This is the simple Fermat primality test. Removing primes, given base 2 this produces the sequence OEIS A001567 <http://oeis.org/A001567>.

is_strong_pseudoprime

  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. This is often called the Miller-Rabin test.

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 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, though in the early 1990s almost all serious software switched to the BPSW test.

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.

miller_rabin_random

  my $maybe_prime = miller_rabin_random($n, 10); # 10 random bases

Takes a positive number ("n") as input and a positive number ("k") of bases to use. Performs "k" Miller-Rabin tests using uniform random bases between 2 and "n-2". This is the correct way to perform "k" Miller-Rabin tests, rather than the common but broken method of using the first "k" primes.

An optional third argument may be given, which is a seed to use. The seed should be a number either in decimal, binary with a leading "0b", hex with a leading "0x", or octal with a leading 0. It will be converted to a GMP integer, so may be large. Typically this is not necessary, but cryptographic applications may prefer the ability to use this, and it allows repeatable test results.

There is no check for duplicate bases. Input sizes below 65-bits make little sense for this function since is_prob_prime is deterministic at that size. For numbers of 65+ bits, the chance of duplicate bases is quite small. The exponentiation approximation for the birthday problem gives a probability of less than 2e-16 for 100 random bases to have a duplicate with a 65-bit input, and less than 2e-35 with a 128-bit input.

is_lucas_pseudoprime

is_strong_lucas_pseudoprime

Takes a positive number as input, and returns 1 if the input is a standard or strong Lucas probable prime. The Selfridge method of choosing D, P, and Q are used (some sources call this a Lucas-Selfridge test). This is one half of the BPSW primality test (the Miller-Rabin strong probable prime test with base 2 being the other half). The canonical BPSW test (page 1401 of Baillie and Wagstaff (1980)) uses the strong Lucas test with Selfridge parameters, but in practice a variety of Lucas tests with different parameters are used by tests calling themselves BPSW.

The standard Lucas test implemented here corresponds to the Lucas test described in FIPS 186-4 section C.3.3, though uses a slightly more efficient calculation. Since the standard Lucas-Selfridge test is a subset of the strong Lucas-Selfridge test, I recommend using the strong test rather than the standard test for cryptographic purposes. It is often slightly faster, has over 4x fewer pseudoprimes, and is the method recommended by Baillie and Wagstaff in their 1980 paper.

is_extra_strong_lucas_pseudoprime

Takes a positive number as input, and returns 1 if the input is an extra-strong Lucas probable prime. This is defined in Grantham (2000), and is a slightly more stringent test than the strong Lucas test, though because different parameters are used the pseudoprimes are not a subset. As expected by the extra conditions, the number of pseudoprimes is less than 2/3 that of the strong Lucas-Selfridge test. Runtime performance is 1.2 to 1.5x faster than the strong Lucas test.

The parameters are selected using the Baillie-OEIS method:

  P = 3;
  Q = 1;
  while ( jacobi( P*P-4, n ) != -1 )
    P += 1;

is_almost_extra_strong_lucas_pseudoprime

Takes a positive number as input and returns 1 if the input is an ``almost'' extra-strong Lucas probable prime. This is the classic extra-strong Lucas test but without calculating the U sequence. This makes it very fast, although as the input increases in size the time converges to the conventional extra-strong implementation: at 30 digits this routine is about 15% faster, at 300 digits it is only 2% faster.

With the current implementations, there is little reason to prefer this unless trying to reproduce specific results. The extra-strong implementation has been optimized to use similar features, removing most of the performance advantage.

An optional second argument (must be between 1 and 256) indicates the increment amount for P parameter selection. The default value of one yields the method described in ``is_extra_strong_lucas_pseudoprime''. A value of 2 yields the method used in Pari <http://pari.math.u-bordeaux.fr/faq.html#primetest>.

Because the "U = 0" condition is ignored, this produces about 5% more pseudoprimes than the extra-strong Lucas test. However this is still only 66% of the number produced by the strong Lucas-Selfridge test. No BPSW counterexamples have been found with any of the Lucas tests described.

is_perrin_pseudoprime

Takes a positive number "n" as input and returns 1 if "n" divides P(n) where P(n) is the Perrin number of "n". The Perrin sequence is defined by

   C<P(0) = 3, P(1) = 0, P(2) = 2;  P(n) = P(n-2) + P(n-3)>

This is not a commonly used test, as it runs 5 to 10 times slower than most of the other probable prime tests and offers little benefit, especially over combined tests like ``is_bpsw_prime'' and ``is_frobenius_underwood_pseudoprime''.

is_frobenius_pseudoprime

Takes a positive number "n" as input, and two optional parameters "a" and "b", and returns 1 if the "n" is a Frobenius probable prime with respect to the polynomial "x^2 - ax + b". Without the parameters, "b = 2" and "a" is the least positive odd number such that "(a^2-4b|n) = -1". This selection has no pseudoprimes below "2^64" and none known. In any case, the discriminant "a^2-4b" must not be a perfect square.

is_frobenius_underwood_pseudoprime

Takes a positive number as input, and returns 1 if the input passes the efficient Frobenius test of Paul Underwood. This selects a parameter "a" as the least non-negative integer such that "(a^2-4|n)=-1", then verifies that "(x+2)^(n+1) = 2a + 5 mod (x^2-ax+1,n)". This combines a Fermat and Lucas test at a computational cost of about 2.5x a strong pseudoprime test. This makes it similar to, but faster than, a Frobenius test.

This test is deterministic (no randomness is used). There are no known pseudoprimes to this test. This test also has no overlap with the BPSW test, making it a very effective method for adding additional certainty.

is_frobenius_khashin_pseudoprime

Takes a positive number as input, and returns 1 if the input passes the Frobenius test of Sergey Khashin. This ensures "n" is not a perfect square, selects the parameter "c" as the smallest odd prime such that "(c|n)=-1", then verifies that "(1+D)^n = (1-D) mod n" where "D = sqrt(c) mod n".

This test is deterministic (no randomness is used). There are no known pseudoprimes to this test.

is_bpsw_prime

Given a positive number input, returns 0 (composite), 2 (definitely prime), or 1 (probably prime), using the BPSW primality test (extra-strong variant).

This function does the extra-strong BPSW test and nothing more. That is, it will skip all pretests and any extra work that the ``is_prob_prime'' test may add.

is_aks_prime

  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 theory, AKS is extremely important. In practice, it is essentially useless. Estimated run time for a 150 digit input is about 9 years, making the case that while the algorithmic complexity growth is polynomial, the constants are ludicrously high. There are some ideas of Bernstein that can reduce this a little, but it would still take years for numbers that ECPP or APR-CL can prove in seconds.

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

is_mersenne_prime

  say "2^607-1 (M607) is a Mersenne prime" if is_mersenne_prime(607);

Takes a positive number "p" as input and returns 1 if "2^p-1" is prime. After some pre-testing, the Lucas-Lehmer test is performed. This is a deterministic unconditional test that runs very fast compared to other primality methods for numbers of comparable size, and vastly faster than any known general-form primality proof methods.

is_llr_prime

Takes a positive number "n" as input and returns one of: 0 (definitely composite), 2 (definitely prime), or -1 (test does not indicate anything). This implements the Lucas-Lehmer-Riesel test for fast deterministic primality testing on numbers of the form "k * 2^n - 1". If "k = 1" then this is a Mersenne number and the Lucas-Lehmer test is used. If the number is not of this form, or if "k <= 2^n", then "-1" will be returned as the test does not apply. Otherwise, the LLR test is performed. While not as fast as the Lucas-Lehmer test for Mersenne numbers, it is almost as fast as a single strong pseudoprime test (i.e. Miller-Rabin test) while giving a certain answer.

is_proth_prime

Takes a positive number "n" as input and returns one of: 0 (definitely composite), 2 (definitely prime), or -1 (test does not indicate anything). This applies Proth's theorem for fast Las Vegas primality testing on numbers of the form "k * 2^n + 1". If the number is not of this form, or if "k <= 2^n", then "-1" will be returned as the test does not apply. Otherwise, a search is performed to find a quadratic nonresidue modulo "n". If none can be found after a brief search, "-1" is returned as no conclusion can be reached. Otherwise, Proth's theorem is checked which conclusively indicates primality. While not as fast as the Lucas-Lehmer test for Mersenne numbers, it is almost as fast as a single strong pseudoprime test (i.e. Miller-Rabin test) while giving a certain answer.

is_miller_prime

  say "$n is definitely prime" if is_miller_prime($n);
  say "$n is definitely prime assuming the GRH" if is_miller_prime($n, 1);

Takes a positive number as input, and returns 1 if the input passes the deterministic Miller test. An optional second argument indicates whether the Generalized Riemann Hypothesis should be assumed, and defaults to 0. Setting the verbose flag to 2 or higher will show how many bases are used. The unconditional test is exponential time, while the conditional test (assuming the GRH) is polynomial time.

This is a very slow method in practice, and generally should not be used. The asymptotic complexity of the GRH version is good in theory, matching ECPP, but in practice it is much slower. The number of bases used by the unconditional test grows quite rapidly, impractically many past about 160 bits, and overflows a 64-bit integer at 456 bits --- sizes that are trivial for the unconditional APR-CL and ECPP tests.

is_nminus1_prime

  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.

is_ecpp_prime

  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.

primes

  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, matching the signature of the function of the same name in Math::Prime::Util.

Values above 64-bit are extra-strong BPSW probable primes.

sieve_primes

  my @primes = sieve_primes(2**100, 2**100 + 10000);
  my @candidates = sieve_primes(2**1000, 2**1000 + 10000, 40000);

Given two arguments "low" and "high", this returns the primes in the interval (inclusive) as a list. It operates similar to primes, though must always have an lower and upper bound and returns a list.

With three arguments "low", "high", and "limit", this does a partial sieve over the inclusive range and returns the list that pass the sieve. If "limit" is less than 2 then it is identical to the two-argument version, in that a primality test will be performed after sieving. Otherwise, sieving is performed up to "limit".

The two-argument version is typically only used internally and adds little functionality. The three-argument version is quite useful for applications that want to apply their own primality or other tests, and wish to have a list of values in the range with no small factors. This is quite common for applications involving prime gaps.

sieve_twin_primes

  my @primes = sieve_twin_primes(2**1000, 2**1000 + 500000);

Given two arguments "low" and "high", this returns each lower twin prime in the interval (inclusive). The result is a list, not a reference.

This does a partial sieve of the range, removes any non-twin candidates, then checks that each pair are both BPSW probable primes. This is substantially more efficient than sieving for all primes followed by removing those that are not twin primes.

sieve_prime_cluster

  # Find some prime septuplets
  my @s = sieve_prime_cluster(2**100, 2**100+1e12, 2,6,8,12,18,20);

Efficiently finds prime clusters between the first two arguments "low" and "high" (inclusive). The remaining arguments describe the cluster. The cluster values must be even, less than 31 bits, and strictly increasing. Given a cluster set "C", the returned values are all primes in the range where "p+c" is prime for all "c" in the cluster set "C".

The cluster is described as offsets from 0, with the implicit prime at 0. Hence an empty list is asking for all primes (the cluster "p+0"). A list with the single value 2 will find all twin primes (the cluster where "p+0" and "p+2" are prime). The list "2,6,8" will find prime quadruplets. Note that there is no requirement that the list denote a constellation (a cluster with minimal distance) --- the list "42,92,606" is just fine.

For long clusters, e.g. OEIS series A213601 <http://oeis.org/A213601> prime 12-tuplets, this will be immensely more efficient than filtering out the cluster from a list of primes. For that example, a range of "10^13" takes less than a second to search --- thousands of times faster than filtering results from primes or twin primes. Shorter clusters are not quite this efficient, and the overhead for returning large arrays should not be ignored.

next_prime

  $n = next_prime($n);

Returns the prime following the input number (the smallest prime number that is greater than the input number). The function ``is_prob_prime'' is used to determine when a prime is found, hence the result is a probable prime (using BPSW).

For large inputs this function is quite a bit faster than GMP's "mpz_nextprime" or Pari's "nextprime".

prev_prime

  $n = prev_prime($n);

Returns the prime preceding the input number (the largest prime number that is less than the input number). 0 is returned if the input is 2 or lower. The function ``is_prob_prime'' is used to determine when a prime is found, hence the result is a probable prime (using BPSW).

lucasu

  say "Fibonacci($_) = ", lucasu(1,-1,$_) for 0..100;

Given integers "P", "Q", and the non-negative integer "k", computes "U_k" for the Lucas sequence defined by "P","Q". These include the Fibonacci numbers ("1,-1"), the Pell numbers ("2,-1"), the Jacobsthal numbers ("1,-2"), the Mersenne numbers ("3,2"), and more.

This corresponds to OpenPFGW's "lucasU" function and gmpy2's "lucasu" function.

lucasv

  say "Lucas($_) = ", lucasv(1,-1,$_) for 0..100;

Given integers "P", "Q", and the non-negative integer "k", computes "V_k" for the Lucas sequence defined by "P","Q". These include the Lucas numbers ("1,-1").

This corresponds to OpenPFGW's "lucasV" function and gmpy2's "lucasv" function.

lucas_sequence

  my($U, $V, $Qk) = lucas_sequence($n, $P, $Q, $k)

Computes "U_k", "V_k", and "Q_k" for the Lucas sequence defined by "P","Q", modulo "n". The modular Lucas sequence is used in a number of primality tests and proofs.

The following conditions must hold:
  - "D = P*P - 4*Q  !=  0"
  - "P > 0"
  - "P < n"
  - "Q < n"
  - "k >= 0"
  - "n >= 2"

primorial

  $p = primorial($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 <http://oeis.org/A034386> and Wikipedia: Primorial definition for natural numbers <http://en.wikipedia.org/wiki/Primorial#Definition_for_natural_numbers>.

pn_primorial

  $p = pn_primorial($n)

Given an unsigned integer argument, returns the product of the first "n" prime numbers. This definition of "p_n#" follows OEIS series A002110 <http://oeis.org/A002110> and Wikipedia: Primorial definition for prime numbers <http://en.wikipedia.org/wiki/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) )

factorial

Given positive integer argument "n", returns the factorial of "n", defined as the product of the integers 1 to "n" with the special case of "factorial(0) = 1". This corresponds to Pari's factorial(n) and Mathematica's "Factorial[n]" functions.

gcd

Given a list of integers, returns the greatest common divisor. This is often used to test for coprimality <https://oeis.org/wiki/Coprimality>.

lcm

Given a list of integers, returns the least common multiple.

gcdext

Given two integers "x" and "y", returns "u,v,d" such that "d = gcd(x,y)" and "u*x + v*y = d". This uses the extended Euclidian algorithm to compute the values satisfying Bézout's Identity.

This corresponds to Pari's "gcdext" function, which was renamed from "bezout" out Pari 2.6. The results will hence match ``bezout'' in Math::Pari.

chinese

  say chinese( [14,643], [254,419], [87,733] );  # 87041638

Solves a system of simultaneous congruences using the Chinese Remainder Theorem (with extension to non-coprime moduli). A list of "[a,n]" pairs are taken as input, each representing an equation "x ≡ a mod n". If no solution exists, "undef" is returned. If a solution is returned, the modulus is equal to the lcm of all the given moduli (see ``lcm''. In the standard case where all values of "n" are coprime, this is just the product. The "n" values must be positive integers, while the "a" values are integers.

vecsum

Returns the sum of all arguments, each of which must be an integer.

vecprod

Returns the product of all arguments, each of which must be an integer.

kronecker

Returns the Kronecker symbol "(a|n)" for two integers. The possible return values with their meanings for odd positive "n" are:

   0   a = 0 mod n
   1   a is a quadratic residue modulo n (a = x^2 mod n for some x)
  -1   a is a quadratic non-residue modulo n

The Kronecker symbol is an extension of the Jacobi symbol to all integer values of "n" from the latter's domain of positive odd values of "n". The Jacobi symbol is itself an extension of the Legendre symbol, which is only defined for odd prime values of "n". This corresponds to Pari's "kronecker(a,n)" function and Mathematica's "KroneckerSymbol[n,m]" function.

binomial

Given integer arguments "n" and "k", returns the binomial coefficient "n*(n-1)*...*(n-k+1)/k!", also known as the choose function. Negative arguments use the Kronenburg extensions <http://arxiv.org/abs/1105.3689/>. This corresponds to Mathematica's "Binomial[n,k]" function, Pari's "binomial(n,k)" function, and GMP's "mpz_bin_ui" function.

For negative arguments, this matches Mathematica. Pari does not implement the "n < 0, k <= n" extension and instead returns 0 for this case. GMP's API does not allow negative "k" but otherwise matches. Math::BigInt does not implement any extensions and the results for "n < 0, k " 0> are undefined.

bernfrac

Returns the Bernoulli number "B_n" for an integer argument "n", as a rational number. Two values are returned, the numerator and denominator. B_1 = 1/2. This corresponds to Pari's bernfrac(n) and Mathematica's "BernoulliB" functions.

harmfrac

Returns the Harmonic number "H_n" for an integer argument "n", as a rational number. Two values are returned, the numerator and denominator. numbers are the sum of reciprocals of the first "n" natural numbers: "1 + 1/2 + 1/3 + ... + 1/n".

harmreal

Returns the Harmonic number "H_n" for an integer argument "n", as a string floating point. An optional second argument indicates the number of digits to be preserved past the decimal place, with a default of 40.

stirling

  say "s(14,2) = ", stirling(14, 2);
  say "S(14,2) = ", stirling(14, 2, 2);

Returns the Stirling numbers of either the first kind (default), the second kind, or the third kind (the unsigned Lah numbers), with the kind selected as an optional third argument. It takes two non-negative integer arguments "n" and "k" plus the optional "type". This corresponds to Pari's "stirling(n,k,{type})" function and Mathematica's "StirlingS1" / "StirlingS2" functions.

Stirling numbers of the first kind are "-1^(n-k)" times the number of permutations of "n" symbols with exactly "k" cycles. Stirling numbers of the second kind are the number of ways to partition a set of "n" elements into "k" non-empty subsets. The Lah numbers are the number of ways to split a set of "n" elements into "k" non-empty lists.

znorder

  $order = znorder(17, "100000000000000000000000065");

Given two positive integers "a" and "n", returns the multiplicative order of "a" modulo "n". This is the smallest positive integer "k" such that "a^k ≡ 1 mod n". Returns 1 if "a = 1". Returns undef if "a = 0" or if "a" and "n" are not coprime, since no value will result in 1 mod n. This corresponds to Pari's "znorder(Mod(a,n))" function and Mathematica's "MultiplicativeOrder[a,n]" function.

znprimroot

Given a positive integer "n", returns the smallest primitive root of "(Z/nZ)^*", or "undef" if no root exists. A root exists when "euler_phi($n) == carmichael_lambda($n)", which will be true for all prime "n" and some composites.

OEIS A033948 <http://oeis.org/A033948> is a sequence of integers where the primitive root exists, while OEIS A046145 <http://oeis.org/A046145> is a list of the smallest primitive roots, which is what this function produces.

sigma

  say "Sum of divisors of $n:", sigma( $n );
  say "sigma_2($n) = ", sigma($n, 2);
  say "Number of divisors: sigma_0($n) = ", sigma($n, 0);

This function takes a positive integer as input and returns the sum of its divisors, including 1 and itself. An optional second argument "k" may be given, which will result in the sum of the "k-th" powers of the divisors to be returned.

This is known as the sigma function (see Hardy and Wright section 16.7, or OEIS A000203). The API is identical to Pari/GP's "sigma" function. This function is useful for calculating things like aliquot sums, abundant numbers, perfect numbers, etc.

ramanujan_tau

Takes a positive integer as input and returns the value of Ramanujan's tau function. The result is a signed integer. This corresponds to Mathematica's "RamanujanTau" function.

valuation

  say "$n is divisible by 2 ", valuation($n,2), " times.";

Given integers "n" and "k", returns the numbers of times "n" is divisible by "k". This is a very limited version of the algebraic valuation meaning, just applied to integers. This corresponds to Pari's "valuation" function. 0 is returned if "n" or "k" is one of the values "-1", 0, or 1.

moebius

  say "$n is square free" if moebius($n) != 0;
  $sum += moebius($_) for (1..200); say "Mertens(200) = $sum";
  say "Mertens(2000) = ", vecsum(moebius(0,2000));

Returns μ(n), the Möbius function (also known as the Moebius, Mobius, or MoebiusMu function) for an integer input. This function is 1 if "n = 1", 0 if "n" is not square free (i.e. "n" has a repeated factor), and "-1^t" if "n" is a product of "t" distinct primes. This is an important function in prime number theory. Like SAGE, we define "moebius(0) = 0" for convenience.

If called with two arguments, they define a range "low" to "high", and the function returns an array with the value of the Möbius function for every n from low to high inclusive.

invmod

  say "The inverse of 42 mod 2017 = ", invmod(42,2017);

Given two integers "a" and "n", return the inverse of "a" modulo "n". If not defined, undef is returned. If defined, then the return value multiplied by "a" equals 1 modulo "n".

consecutive_integer_lcm

  $lcm = consecutive_integer_lcm($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 factorials.

partitions

Calculates the partition function p(n) for a non-negative integer input. This is the number of ways of writing the integer n as a sum of positive integers, without restrictions. This corresponds to Pari's "numbpart" function and Mathematica's "PartitionsP" function. The values produced in order are OEIS series A000041 <http://oeis.org/A000041>.

This uses a combinatorial calculation, which means it will not be very fast compared to Pari, Mathematica, or FLINT which use the Rademacher formula using multi-precision floating point. In 10 seconds, the pure Perl version can produce "partitions(10_000)" while with Math::Prime::Util::GMP it can do "partitions(220_000)". In contrast, in about 10 seconds Pari can solve "numbpart(22_000_000)".

If you want the enumerated partitions, see Integer::Partition. It is very fast and uses an extremely memory efficient iterator. It is not, however, practical for producing the partition number for values over 100 or so.

Pi

Takes a positive integer argument "n" and returns the constant Pi with that many digits (including the leading 3). Rounding is performed.

The implementation uses AGM and is only slightly slower than MPFR (which has tighter bounds on the intermediate bits and exit conditions).

exp_mangoldt

  say "exp(lambda($_)) = ", exp_mangoldt($_) for 1 .. 100;

Returns EXP(Λ(n)), the exponential of the Mangoldt function (also known as von Mangoldt's function) for an integer value. The Mangoldt function is equal to log p if n is prime or a power of a prime, and 0 otherwise. We return the exponential so all results are integers. Hence the return value for "exp_mangoldt" is:

   p   if n = p^m for some prime p and integer m >= 1
   1   otherwise.

totient

  say "The Euler totient of $n is ", totient($n);

Returns φ(n), the Euler totient function (also called Euler's phi or phi function) for an integer value. This is an arithmetic function which counts the number of positive integers less than or equal to "n" that are relatively prime to "n". Given the definition used, "totient" will return 0 for all "n < 1". This follows the logic used by SAGE. Mathematica and Pari return "totient(-n)" for "n < 0". Mathematica returns 0 for "n = 0", Pari pre-2.6.2 raises and exception, and Pari 2.6.2 and newer returns 2.

jordan_totient

  say "Jordan's totient J_$k($n) is ", jordan_totient($k, $n);

Returns Jordan's totient function for a given integer value. Jordan's totient is a generalization of Euler's totient, where
"jordan_totient(1,$n) == euler_totient($n)" This counts the number of k-tuples less than or equal to n that form a coprime tuple with n. As with "totient", 0 is returned for all "n < 1". This function can be used to generate some other useful functions, such as the Dedekind psi function, where "psi(n) = J(2,n) / J(1,n)".

carmichael_lambda

Returns the Carmichael function (also called the reduced totient function, or Carmichael λ(n)) of a positive integer argument. It is the smallest positive integer "m" such that "a^m = 1 mod n" for every integer "a" coprime to "n". This is OEIS series A002322 <http://oeis.org/A002322>.

liouville

Returns λ(n), the Liouville function for a non-negative integer input. This is -1 raised to Ω(n) (the total number of prime factors).

is_power

  say "$n is a perfect square" if is_power($n, 2);
  say "$n is a perfect cube" if is_power($n, 3);
  say "$n is a ", is_power($n), "-th power";

Given a single positive integer input "n", returns k if "n = p^k" for some integer "p > 1, k > 1", and 0 otherwise. The k returned is the largest possible. This can be used in a boolean statement to determine if "n" is a perfect power.

If given two arguments "n" and "k", returns 1 if "n" is a "k-th" power, and 0 otherwise. For example, if "k=2" then this detects perfect squares.

This corresponds to Pari/GP's "ispower" function, with the limitations of only integer arguments and no third argument may be given to return the root.

factor

  @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 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 <http://sourceforge.net/projects/yafu/>, msieve <http://sourceforge.net/projects/msieve/>, gmp-ecm <http://ecm.gforge.inria.fr/>, and GGNFS <http://sourceforge.net/projects/ggnfs/>.

trial_factor

  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). For larger numbers, faster methods for complete factoring have been known since the 17th century.

For inputs larger than about 1000 digits, a dynamic product/remainder tree is used, which is faster than GMP's native methods. This helps when pruning composites or looking for very small factors.

prho_factor

  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. Typically ``pbrent_factor'' will be preferred as it behaves similarly but runs quite a bit faster. They use different parameters however, so are not completely identical.

pbrent_factor

  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, delayed "gcd" computations, 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.

pminus1_factor

  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 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 <http://sourceforge.net/projects/yafu/>, and nowhere close to the speed of the version included with modern GMP-ECM with large B values (it is actually quite a bit faster than GMP-ECM with small smoothness values).

pplus1_factor

  my @factors = pplus1_factor($n);

Given a positive number input, tries to discover a factor using Williams' "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 the first stage. 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.

holf_factor

  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 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.

squfof_factor

  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 indicates it is generally slower than the "prho" and "pbrent" implementations.

ecm_factor

  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.

qs_factor

  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.

REFERENCES

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>
Jon Grantham, "Frobenius Pseudoprimes", Mathematics of Computation, v70 n234, March 2000, pp 873-891. <http://www.ams.org/journals/mcom/2001-70-234/S0025-5718-00-01197-2/>
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. <http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.33.4409>

AUTHORS

Dana Jacobsen <[email protected]>

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

ACKNOWLEDGEMENTS

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 do it 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

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