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Re: Re: Re: Factors

by gumby (Scribe)
on Jun 13, 2002 at 15:43 UTC ( #174242=note: print w/ replies, xml ) Need Help??


in reply to Re: Re: Factors
in thread Vampire Numbers

If you want to find a square root of a (mod p) use the Shanks-Tonelli algorithm.

# Calculate the least non-negative remainder when an integer a # is divided by a positive integer b. sub mod { my ($a, $b) = @_; my $c = $a % $b; return $c if ($a >= 0); return 0 if ($c == 0); return ($c + $b); } # Calculate a^b (mod c), where a,b and c are integers and a,b>=0, c>=1 + sub mpow { my ($a, $b, $c) = @_; my ($x, $y, $z) = ($a, $b, 1); while ($y > 0) { while ($y % 2 == 0) { $y = $y / 2; $x = $x**2 % $c; } $y--; $z = mod($z * $x, $c); } return $z; } # Shanks-Tonelli algorithm to calculate y^2 = a (mod p) for p an odd p +rime sub tonelli { my ($a, $p) = @_; my ($b, $e, $g, $h, $i, $m, $n, $q, $r, $s, $t, $x, $y, $z); $q = $p - 1; $t = mpow($a, $q/2, $p); return 0 if ($t = $q); # a is a quadratic non-residue mod p $s = $q; $e = 0; while ($s % 2 == 0) { $s = $s/2; $e++; } # p-1 = s * 2^e; $x = mpow($a, ++$s/2, $p); $b = mpow($a, $s, $p); return $x if ($b == 1); $n = 2; RES: while ($n >= 2) { $t = mpow($n, $q/2, $p); last RES if ($t == $q); } continue { $n++; } # n is the least quadratic non-residue mod p $g = mpow($n, $s, $p); $r = $e; OUT: { do { $y = $b; $m = 0; IN: while ($m <= $r-1) { last IN if ($y == 1); $y = $y**2 % $p; } continue { $m++; } return $x if ($m == 0); $z = $r - $m - 1; $h = $g; for ($i = 1; $i < $z; $i++) {$h = $h**2 % $p;} $x = ($x * $h) % $p; $b = ($b * $h**2) % $p; $g = $h**2 % $p; $r = $m; } while 1; } }


Comment on Re: Re: Re: Factors
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Re: Re: Re: Re: Factors
by gumby (Scribe) on Jun 15, 2002 at 19:21 UTC
    Here's a more 'reference' style implementation. Basically, this is the algorithm you'll find in the paper 'Square Roots, From 1; 24, 51, 10 To Dan Shanks' by Ezra Brown, but at the cost of using Math::BigInt.
    #!/usr/bin/perl -w use Math::BigInt ':constant'; use strict; sub tonelli { my ($s, $e, $n, $x, $b, $g, $r, $m, $t); my ($a, $p) = @_; die "$a has no square roots (mod $p)" if $a**(($p-1)/2) % $p == -1; $s = $p-1; $e = 0; while ($s % 2 == 0) { $s = $s / 2; $e++; } for ($n = 2; $n >= 2; $n++) { last if $n**(($p-1)/2) % $p == $p-1; } $x = $a**(($s+1)/2) % $p; $b = $a**$s % $p; $g = $n**$s % $p; $r = $e; $t = $b; while (1) { for ($m = 0; $m <= $r-1; $m++) { last if $t % $p == 1; $t = $t**2; } return $x if $m == 0; $x = $x * $g**(2*($r-$m-1)) % $p; $b = $b * $g**(2*($r-$m-1)) % $p; $g = $g**(2*($r-$m)) % $p; $r = $m; } }

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