Tonelli-Shanks algorithm: Difference between revisions

=={{header|Perl 6}}==

{{works with|Rakudo|2016.10}}

Translation of the Wikipedia pseudocode, heavily influenced by Sidef and Python.

<lang perl6># Legendre operator (𝑛│𝑝)

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

given 𝑛.expmod( (𝑝-1) div 2, 𝑝 ) {

when 0 { 0 }

when 1 { 1 }

default { -1 }

}

}

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;

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;

my $b;

while (($𝑡-1) % 𝑝) {

my $𝑡2 = $𝑡² % 𝑝;

for 1 .. $𝑀 {

if ($𝑡2-1) %% 𝑝 {

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

$𝑀 = $_;

last;

}

$𝑡2 = $𝑡2² % 𝑝;

}

$𝑅 = ($𝑅 * $b) % 𝑝;

$𝑐 = $b² % 𝑝;

$𝑡 = ($𝑡 * $𝑐) % 𝑝;

}

$𝑅;

}

my @tests = (

(10, 13),

(56, 101),

(1030, 10009),

(44402, 100049),

(665820697, 1000000009),

(881398088036, 1000000000039),

(41660815127637347468140745042827704103445750172002,

100000000000000000000000000000000000000000000000577)

);

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

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

die if ($t² - $n) % $p;

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

}</lang>

{{out}}

<pre>Roots of 10 are (7, 6) mod 13

Roots of 56 are (37, 64) mod 101

Roots of 1030 are (1632, 8377) mod 10009

Roots of 44402 are (30468, 69581) mod 100049

Roots of 665820697 are (378633312, 621366697) mod 1000000009

Roots of 881398088036 are (791399408049, 208600591990) mod 1000000000039

Roots of 41660815127637347468140745042827704103445750172002 are (32102985369940620849741983987300038903725266634508, 67897014630059379150258016012699961096274733366069) mod 100000000000000000000000000000000000000000000000577

</pre>