lucas-miller_factorization_method.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 07 January 2020
# https://github.com/trizen

# A simple factorization method, using the Lucas `U_n(P,Q)` sequences.
# Inspired by the Miller-Rabin factorization method.

# Works best on Lucas pseudoprimes.

# See also:
#   https://en.wikipedia.org/wiki/Lucas_pseudoprime
#   https://en.wikipedia.org/wiki/Miller-Rabin_primality_test

func lucas_miller_factor(n, j=1, k=100) {

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

    for (1..k) {

        var P = min(irand(1, 1e6), irand(1, n-1))
        var Q = min(irand(1, 1e6), irand(1, n-1))*[1,-1].rand

        next if is_square(P*P - 4*Q)

        var T     = powmod(Q, d, n)
        var (U,V) = lucasUVmod(P, Q, d, n)

        for b in (0..r) {

            for g in (gcd(U,n), gcd(V,n), gcd(V-P,n)) {
                if (g.is_between(2, n-1)) {
                    return g
                }
            }

            U = mulmod(U, V, n)
            V = mulmod(V, V, n)
            V = submod(V, 2*T, n)
            T = mulmod(T, T, n)
        }
    }

    return 1
}

say lucas_miller_factor(16641689036184776955112478816668559)
say lucas_miller_factor(17350074279723825442829581112345759)
say lucas_miller_factor(61881629277526932459093227009982733523969186747)
say lucas_miller_factor(122738580838512721992324860157572874494433031849, -1)
say lucas_miller_factor(181490268975016506576033519670430436718066889008242598463521)
say lucas_miller_factor(173315617708997561998574166143524347111328490824959334367069087)
say lucas_miller_factor(57981220983721718930050466285761618141354457135475808219583649146881)
say lucas_miller_factor(2425361208749736840354501506901183117777758034612345610725789878400467)
say lucas_miller_factor(131754870930495356465893439278330079857810087607720627102926770417203664110488210785830750894645370240615968198960237761)