Tonelli-Shanks algorithm: Difference between revisions

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<lang perl6># Legendre operator (𝑛│𝑝)

sub infix:<│> (Int \𝑛, Int \𝑝 where (𝑝.is-prime ~~and~~ ?(𝑝 != 2))) {

sub infix:<│> (Int \𝑛, Int \𝑝 where 𝑝.is-prime && (𝑝 != 2)) {

given 𝑛.expmod( (𝑝-1) div 2, 𝑝 ) {

when 0 { 0 }

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}

sub tonelli-shanks ( \𝑛, \𝑝 where ((𝑛│𝑝) > 0 )) {

sub tonelli-shanks ( \𝑛, \𝑝 where (𝑛│𝑝) > 0 ) {

my $𝑄 = 𝑝 - 1;

my $𝑆 = 0;

$𝑄 +>= 1 and $𝑆++ while $𝑄 %% 2;

return 𝑛.expmod((𝑝+1) div 4, 𝑝) if $𝑆 == 1;

my $𝑐 = ((2..𝑝).first: (*│𝑝) < 0).expmod($𝑄, 𝑝);

my $𝑐;

for 2 .. 𝑝 {

$𝑐 = .expmod($𝑄, 𝑝) and last if ($_│𝑝) < 0;

}

my $𝑅 = 𝑛.expmod( ($𝑄+1) +> 1, 𝑝 );

my $𝑡 = 𝑛.expmod( $𝑄, 𝑝 );

my $𝑀 = ~~$𝑆;~~

while ($𝑡-1) % 𝑝 {

my $b;

while (($𝑡-1) % 𝑝) {

my $𝑡2 = $𝑡² % 𝑝;

for 1 .. $𝑀 {

for 1 .. $𝑆 {

if ($𝑡2-1) %% 𝑝 {

$b = $𝑐.expmod( 1 +< ( $𝑀 - 1 - $_ ), 𝑝 );

$b = $𝑐.expmod(1 +< ($𝑆-1-$_), 𝑝);

$𝑀 = $_;

$𝑆 = $_;

last;

}

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(56, 101),

(1030, 10009),

(1032, 10009),

(44402, 100049),

(665820697, 1000000009),

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for @tests -> ($n, $p) {

my $t = tonelli-shanks($n, $p);

try my $t = tonelli-shanks($n, $p);

~~die~~ if ($t² - $n) % $p;

say "No solution for ({$n}, {$p})." and next if !$t or ($t² - $n) % $p;

say "Roots of $n are ($t, {$p-$t}) mod $p";

}</lang>

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Roots of 56 are (37, 64) mod 101

Roots of 1030 are (1632, 8377) mod 10009

No solution for (1032, 10009).

Roots of 44402 are (30468, 69581) mod 100049

Roots of 665820697 are (378633312, 621366697) mod 1000000009