divisor_of_p-1_factorization_method.sf

#!/usr/bin/ruby

# Author: Daniel "Trizen" Șuteu
# Date: 27 February 2022
# https://github.com/trizen

# A new special-purpose integer factorization method, finding a factor of n
# if a large enough divisor of p-1 is known, where p is a prime dividing n.

func dpm1_factor(n, pm1_divisor = 1, reps = 1e3) {

    for k in (1..reps) {

        var a = pm1_divisor*k
        var u = idiv(n, a)

        break if (u <= 1)

        bsearch_le(2, u, {|b|

            #var x = idiv(isqrt(a*a + 4*a*b*n - 2*a*b + b*b) - a - b, 2*a*b)
            var (x) = iquadratic_formula(a*b, a+b, 1-n)

            var t = ((x*a + 1) * (x*b + 1))
            var g = gcd(t, n)

            if (g.is_between(2, n-1)) {
                say "[#{k} tries] Found factor: #{g} with a,b = [#{a}, #{b}] and x = #{x}"
                return g
            }

            break if (n/t < 1.001)     # optimization

            n <=> t
        })
    }

    return 1
}

dpm1_factor(503*863, 2)                                                  #=> 503
dpm1_factor(2**64 + 1, 256)                                              #=> 274177
dpm1_factor(2**128 + 1, 116503103764643)                                 #=> 59649589127497217
dpm1_factor((114*(2**127 - 1) + 1) * random_prime(1e50), 2**127 - 1)     #=> 19396094914493492417412352623610788052879

__END__
[36 tries] Found factor: 503 with a,b = [72, 1508] and x = 1
[17 tries] Found factor: 274177 with a,b = [4352, 1034834473201] and x = 63
[256 tries] Found factor: 59649589127497217 with a,b = [29824794563748608, 1426172300171282287590] and x = 2
[38 tries] Found factor: 19396094914493492417412352623610788052879 with a,b = [6465364971497830805804117541203596017626, 4102844459326514132014637137633862970170675382519] and x = 3