miller-rabin_factorization_method.sf

#!/usr/bin/ruby

# Factorization method, based on the Miller-Rabin primality test.
# Described in the book "Elementary Number Theory", by Peter Hackman.

# Works best on Carmichael numbers.

# Example:
#   N   = 1729
#   N-1 = 2^6 * 27

# Then, we find that:
#       2^(2*27) == 1065 != -1 (mod N)
# and
#       2^(4*27) == 1 (mod N)

# This proves that N is composite and gives the following factorization:
#   x = 2^(2*27) (mod N)
#   N = gcd(x+1, N) * gcd(x-1, N)
#   N = gcd(1065+1, N) * gcd(1065-1, N)
#   N = 13 * 133

# See also:
#   https://www.math.waikato.ac.nz/~kab/509/bigbook.pdf
#   https://en.wikipedia.org/wiki/Miller-Rabin_primality_test

func miller_rabin_factor(n, tries=100) {

    var D = n-1
    var s = D.valuation(2)
    var r = s-1
    var d = D>>s

    tries.times {

        var a = random_prime(1e7)
        var x = powmod(a, d, n)

        for b in (0..r) {
            break if ((x == 1) || (x == D))
            for i in (1, -1) {
                var g = gcd(x+i, n)
                if (g.is_between(2, n-1)) {
                    return g
                }
            }
            x = powmod(x, 2, n)
        }
    }

    return 1
}

say miller_rabin_factor(1729)
say miller_rabin_factor(335603208601)
say miller_rabin_factor(30459888232201)
say miller_rabin_factor(162021627721801)
say miller_rabin_factor(1372144392322327801)
say miller_rabin_factor(7520940423059310542039581)
say miller_rabin_factor(8325544586081174440728309072452661246289)
say miller_rabin_factor(181490268975016506576033519670430436718066889008242598463521)
say miller_rabin_factor(57981220983721718930050466285761618141354457135475808219583649146881)
say miller_rabin_factor(131754870930495356465893439278330079857810087607720627102926770417203664110488210785830750894645370240615968198960237761)