As hinted by Abigail-II, exponentiating mod a number is better accomplished by keeping the numbers reduced at every step. The straightforward algorithm would be 
# modexp ( base, exp, n ) = base**exp mod n
sub modexp {
    my ($base, $exp, $n) = @_;
    die "Negative exponent" if $exp<0;
    my $res = 1;
    while ($exp>0) {
    $res = ( $res * $base ) % $n;
    $exp--;
    }
    return $res;
}

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This has the disadvantage of being of linear complexity in the exponent. An algorithm linear in log($exp) is as follows:

sub binmodexp {
    my ($base, $exp, $n) = @_;
    die "Negative exponent" if $exp<0;
    my $res = 1;
    my $mul = $base % $n;
    while ($exp) {
    if ($exp & 1) {
        $res = ( $res * $mul ) % $n;
    }
    $exp>>=1;
    $mul = ($mul*$mul) % $n;
    }
    return $res;
}

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It is based on the simple (but powerful) observation that if you have to take, say, the 37-th power of a number a, as 37 = 2^5 + 2^2 + 1, it is enough to multiply the factors 

a,
a^(2^2) = (a^2) ^ 2
and
a^(2^5) = ((((a^2) ^ 2) ^ 2) ^ 2) ^ 2


Best regards

Antonio Bellezza

The stupider the astronaut, the easier it is to win the trip to Vega - A. Tucket

In reply to Re: The Exponentiation of (not-so) Large Primes by abell
in thread The Exponentiation of (not-so) Large Primes by titivillus

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