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is_bfsw_pseudoprime.pl
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is_bfsw_pseudoprime.pl
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#!/usr/bin/perl
# Author: Daniel "Trizen" Șuteu
# Date: 31 October 2023
# https://github.com/trizen
# A new primality test, using only the Lucas V sequence.
# This test is a simplification of the strengthen BPSW test:
# https://arxiv.org/abs/2006.14425
use 5.036;
use Math::GMPz;
use constant {
USE_METHOD_A_STAR => 0, # true to use the A* method in finding (P,Q)
};
sub check_lucasV ($P, $Q, $m) {
state $t = Math::GMPz::Rmpz_init();
Math::GMPz::Rmpz_add_ui($t, $m, 1);
my $s = Math::GMPz::Rmpz_scan1($t, 0);
Math::GMPz::Rmpz_div_2exp($t, $t, $s + 1);
my $V1 = Math::GMPz::Rmpz_init_set_ui(2);
my $V2 = Math::GMPz::Rmpz_init_set_ui($P);
my $Q1 = Math::GMPz::Rmpz_init_set_ui(1);
my $Q2 = 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, $m);
if ($bit) {
Math::GMPz::Rmpz_mul_si($Q2, $Q1, $Q);
Math::GMPz::Rmpz_mul($V1, $V1, $V2);
Math::GMPz::Rmpz_powm_ui($V2, $V2, 2, $m);
Math::GMPz::Rmpz_submul_ui($V1, $Q1, $P);
Math::GMPz::Rmpz_submul_ui($V2, $Q2, 2);
Math::GMPz::Rmpz_mod($V1, $V1, $m);
}
else {
Math::GMPz::Rmpz_set($Q2, $Q1);
Math::GMPz::Rmpz_mul($V2, $V2, $V1);
Math::GMPz::Rmpz_powm_ui($V1, $V1, 2, $m);
Math::GMPz::Rmpz_submul_ui($V2, $Q1, $P);
Math::GMPz::Rmpz_submul_ui($V1, $Q2, 2);
Math::GMPz::Rmpz_mod($V2, $V2, $m);
}
}
Math::GMPz::Rmpz_mul($Q1, $Q1, $Q2);
Math::GMPz::Rmpz_mod($Q1, $Q1, $m);
Math::GMPz::Rmpz_mul_si($Q2, $Q1, $Q);
Math::GMPz::Rmpz_mul($V1, $V1, $V2);
Math::GMPz::Rmpz_submul_ui($V1, $Q1, $P);
Math::GMPz::Rmpz_mul($Q2, $Q2, $Q1);
for (1 .. $s) {
Math::GMPz::Rmpz_powm_ui($V1, $V1, 2, $m);
Math::GMPz::Rmpz_submul_ui($V1, $Q2, 2);
Math::GMPz::Rmpz_powm_ui($Q2, $Q2, 2, $m);
}
Math::GMPz::Rmpz_mod($V1, $V1, $m);
Math::GMPz::Rmpz_set_si($t, 2 * $Q);
Math::GMPz::Rmpz_congruent_p($V1, $t, $m) || return 0;
Math::GMPz::Rmpz_set_si($t, $Q * $Q);
Math::GMPz::Rmpz_congruent_p($Q2, $t, $m) || return 0;
return 1;
}
sub findQ ($n) {
for (my $k = 2 ; ; ++$k) {
my $D = (-1)**$k * (2 * $k + 1);
my $K = Math::GMPz::Rmpz_si_kronecker($D, $n);
if ($K == -1) {
return ((1 - $D) / 4);
}
elsif ($K == 0 and abs($D) < $n) {
return undef;
}
elsif ($k == 20 and Math::GMPz::Rmpz_perfect_square_p($n)) {
return undef;
}
}
}
sub findP ($n, $Q) {
for (my $P = 2 ; ; ++$P) {
my $D = $P * $P - 4 * $Q;
my $K = Math::GMPz::Rmpz_si_kronecker($D, $n);
if ($K == -1) {
return $P;
}
elsif ($K == 0 and abs($D) < $n) {
return undef;
}
elsif ($P == 20 and Math::GMPz::Rmpz_perfect_square_p($n)) {
return undef;
}
}
}
sub is_bfsw_psp ($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);
my ($P, $Q);
if (USE_METHOD_A_STAR) {
$P = 1;
$Q = findQ($n) // return 0;
if ($Q == -1) {
$P = 5;
$Q = 5;
}
}
else {
$Q = -2;
$P = findP($n, $Q) // return 0;
}
check_lucasV($P, $Q, $n);
}
my @strong_lucas_psp = (
5459, 5777, 10877, 16109, 18971, 22499, 24569, 25199, 40309, 58519, 75077, 97439,
100127, 113573, 115639, 130139, 155819, 158399, 161027, 162133, 176399, 176471, 189419, 192509,
197801, 224369, 230691, 231703, 243629, 253259, 268349, 288919, 313499, 324899
);
my @extra_strong_lucas_psp = (
989, 3239, 5777, 10877, 27971, 29681, 30739, 31631, 39059, 72389, 73919, 75077,
100127, 113573, 125249, 137549, 137801, 153931, 155819, 161027, 162133, 189419, 218321, 231703,
249331, 370229, 429479, 430127, 459191, 473891, 480689, 600059, 621781, 632249, 635627
);
foreach my $n (913, 150267335403, 430558874533, 14760229232131, 936916995253453, @strong_lucas_psp, @extra_strong_lucas_psp) {
if (is_bfsw_psp($n)) {
say "Counter-example: $n";
}
}
use ntheory qw(is_prime);
my $from = 1;
my $to = 1e5;
my $count = 0;
foreach my $n ($from .. $to) {
if (is_bfsw_psp($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.";
is_bfsw_psp(3 * Math::GMPz->new("2")**5134 - 1) or die "error";
is_bfsw_psp(Math::GMPz->new(10)**2000 + 4561) or die "error";
__END__
Inspired by the paper "Strengthening the Baillie-PSW primality test", I propose a simplified test based on Lucas V-pseudoprimes, that requires computing only the Lucas V sequence, making it faster than the full BPSW test, while being about as strong.
The first observation was that none of the 5 vpsp terms < 10^15 satisfy:
Q^(n+1) == Q^2 (mod n)
This gives us a simple test:
V_{n+1}(P,Q) == 2*Q (mod n)
Q^(n+1) == Q^2 (mod n)
where (P,Q) are selected using Method A*.