BPSW_primality_test_mpz.pl

#!/usr/bin/perl

# The Baillie-PSW primality test, named after Robert Baillie, Carl Pomerance, John Selfridge, and Samuel Wagstaff.

# No counter-examples are known to this test.

# Algorithm: given an odd integer n, that is not a perfect power:
#   1. Perform a (strong) base-2 Fermat test.
#   2. Find the first D in the sequence 5, −7, 9, −11, 13, −15, ... for which the Jacobi symbol (D/n) is −1.
#      Set P = 1 and Q = (1 − D) / 4.
#   3. Perform a strong Lucas probable prime test on n using parameters D, P, and Q.

# See also:
#   https://en.wikipedia.org/wiki/Lucas_pseudoprime
#   https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test

use 5.020;
use warnings;
use experimental qw(signatures);

use Math::GMPz;

sub findQ ($n) {
    for (my $k = 2 ; ; ++$k) {
        my $D = (-1)**$k * (2 * $k + 1);

        if (Math::GMPz::Rmpz_si_kronecker($D, $n) == -1) {
            return ((1 - $D) / 4);
        }
    }
}

sub BPSW_primality_test ($n) {

    $n = Math::GMPz::Rmpz_init_set_str($n, 10) if ref($n) ne 'Math::GMPz';

    return 0 if Math::GMPz::Rmpz_cmp_ui($n, 1) <= 0;
    return 1 if Math::GMPz::Rmpz_cmp_ui($n, 2) == 0;

    return 0 if Math::GMPz::Rmpz_even_p($n);
    return 0 if Math::GMPz::Rmpz_perfect_power_p($n);

    state $d = Math::GMPz::Rmpz_init_nobless();
    state $t = Math::GMPz::Rmpz_init_nobless();

    Math::GMPz::Rmpz_set_ui($t, 2);

    # Fermat base-2 test (a strong Miller-Rabin test should be preferred instead)
    Math::GMPz::Rmpz_sub_ui($d, $n, 1);
    Math::GMPz::Rmpz_powm($t, $t, $d, $n);
    Math::GMPz::Rmpz_cmp_ui($t, 1) and return 0;

    my $P = 1;
    my $Q = findQ($n);

    Math::GMPz::Rmpz_add_ui($d, $d, 2);                 # d = n+1
    my $s = Math::GMPz::Rmpz_scan1($d, 0);              # s = valuation(n, 2)
    Math::GMPz::Rmpz_div_2exp($t, $d, $s+1);            # t = d >> (s+1)

    my $U1 = Math::GMPz::Rmpz_init_set_ui(1);
    my ($V1, $V2) = (Math::GMPz::Rmpz_init_set_ui(2), Math::GMPz::Rmpz_init_set_ui($P));
    my ($Q1, $Q2) = (Math::GMPz::Rmpz_init_set_ui(1), Math::GMPz::Rmpz_init_set_ui(1));

    foreach my $bit (split(//, Math::GMPz::Rmpz_get_str($t, 2))) {

        Math::GMPz::Rmpz_mul($Q1, $Q1, $Q2);
        Math::GMPz::Rmpz_mod($Q1, $Q1, $n);

        if ($bit) {
            Math::GMPz::Rmpz_mul_si($Q2, $Q1, $Q);
            Math::GMPz::Rmpz_mul($U1, $U1, $V2);
            Math::GMPz::Rmpz_mul($V1, $V1, $V2);

            Math::GMPz::Rmpz_powm_ui($V2, $V2, 2, $n);
            Math::GMPz::Rmpz_sub($V1, $V1, $Q1);
            Math::GMPz::Rmpz_submul_ui($V2, $Q2, 2);

            Math::GMPz::Rmpz_mod($V1, $V1, $n);
            Math::GMPz::Rmpz_mod($U1, $U1, $n);
        }
        else {
            Math::GMPz::Rmpz_set($Q2, $Q1);
            Math::GMPz::Rmpz_mul($U1, $U1, $V1);
            Math::GMPz::Rmpz_mul($V2, $V2, $V1);
            Math::GMPz::Rmpz_sub($U1, $U1, $Q1);

            Math::GMPz::Rmpz_powm_ui($V1, $V1, 2, $n);
            Math::GMPz::Rmpz_sub($V2, $V2, $Q1);
            Math::GMPz::Rmpz_submul_ui($V1, $Q2, 2);

            Math::GMPz::Rmpz_mod($V2, $V2, $n);
            Math::GMPz::Rmpz_mod($U1, $U1, $n);
        }
    }

    Math::GMPz::Rmpz_mul($Q1, $Q1, $Q2);
    Math::GMPz::Rmpz_mul_si($Q2, $Q1, $Q);
    Math::GMPz::Rmpz_mul($U1, $U1, $V1);
    Math::GMPz::Rmpz_mul($V1, $V1, $V2);
    Math::GMPz::Rmpz_sub($U1, $U1, $Q1);
    Math::GMPz::Rmpz_sub($V1, $V1, $Q1);
    Math::GMPz::Rmpz_mul($Q1, $Q1, $Q2);

    if (Math::GMPz::Rmpz_divisible_p($U1, $n)) {
        return 1;
    }

    if (Math::GMPz::Rmpz_divisible_p($V1, $n)) {
        return 1;
    }

    for (1 .. $s-1) {

        Math::GMPz::Rmpz_powm_ui($V1, $V1, 2, $n);
        Math::GMPz::Rmpz_submul_ui($V1, $Q1, 2);
        Math::GMPz::Rmpz_powm_ui($Q1, $Q1, 2, $n);

        if (Math::GMPz::Rmpz_divisible_p($V1, $n)) {
            return 1;
        }
    }

    return 0;
}

#
## Run some tests
#

use ntheory qw(is_prime);

my $from  = 1;
my $to    = 1e5;
my $count = 0;

foreach my $n ($from .. $to) {
    if (BPSW_primality_test($n)) {
        if (not is_prime($n)) {
            say "Counter-example: $n";
        }
        ++$count;
    }
    elsif (is_prime($n)) {
        say "Missed a prime: $n";
    }
}

say "There are $count primes between $from and $to.";