lucas_factorization_method_generalized.sf

#!/usr/bin/ruby

# Author: Daniel "Trizen" Șuteu
# Date: 17 May 2019
# https://github.com/trizen

# A new integer factorization method, using the modular Lucas U sequence.

# It uses the smallest divisor `d` of `p - kronecker(D, n)`, such that `U_d(P,Q) = 0 (mod p)`.

# By selecting a small bound B, we compute `k = lcm(1..B)`, hoping that `k` is a
# multiple of `d`, then `gcd(U_k(P,Q) (mod n), n)` in a non-trivial factor of `n`.

# This method is similar in flavor to Pollard's p-1 and Williams's p+1 methods.

func lucas_factorization(n, B = n.ilog2**2, a = 1, b = 5) {

    var L = B.consecutive_lcm

    for P in (a .. b) {           # P > 0, P < n

        var Q = ([1,-1].rand * irand(min(1e7, n-1)))
        var D = (P*P - 4*Q)

        D || next

        var F = lucasUmod(P, Q, L, n)
        var g = gcd(F, n)

        if (g.is_between(2, n-1)) {
            return g
        }
    }

    return 1
}

say lucas_factorization(257221 * 470783,                700);    #=> 470783           (p+1 is  700-smooth)
say lucas_factorization(333732865481 * 1632480277613,  3000);    #=> 333732865481     (p-1 is 3000-smooth)
say lucas_factorization(1124075136413 * 3556516507813, 4000);    #=> 1124075136413    (p+1 is 4000-smooth)
say lucas_factorization(6555457852399 * 7864885571993,  700);    #=> 6555457852399    (p-1 is  700-smooth)
say lucas_factorization(7553377229 * 588103349,         800);    #=> 7553377229       (p+1 is  800-smooth)

for k in (10..50) {

    var n = (random_prime(2**k) * random_prime(2**k))
    var B = 8*int(exp(sqrt(log(n) * log(log(n))) / 2))
    var p = lucas_factorization(n, B)

    if (p.is_prime) {
        say "#{n} = #{p} * #{n/p}"
    }
}

__END__
1098947 = 683 * 1609
3059429 = 1483 * 2063
6512993 = 821 * 7933
57606617 = 5431 * 10607
174907669 = 7937 * 22037
1759396517 = 46199 * 38083
2378045827 = 76679 * 31013
9236684929 = 41491 * 222619
26180302453 = 108967 * 240259
17777879953 = 98323 * 180811
2692088381837 = 1374257 * 1958941
2335955746093 = 4312961 * 541613
90707847818641 = 12850811 * 7058531
472153976836973 = 14571107 * 32403439
334066509182312441 = 1043357477 * 320184133
633497402663384509 = 876274397 * 722944097
9284234147413737557 = 2673658063 * 3472483739
158495270425670543821 = 14872678039 * 10656807739
167821505065613669143 = 10577372321 * 15866086583
211225651750010050357 = 4448313713 * 47484432389
11854320114182178323269 = 99693495271 * 118907658739
772323021695922630289 = 54803533259 * 14092577171
168233019237186593148037 = 487441215403 * 345134990479
49758229648443541997314349 = 7016837554687 * 7091261449427
6384031017948052159207031 = 2300064692551 * 2775587590481