difference_of_matrices_factorization_method.sf

#!/usr/bin/ruby

# Author: Daniel "Trizen" Șuteu
# Date: 13 March 2021
# https://github.com/trizen

# A new factorization method, using Fibonacci-like matrices.

# The idea is to try to find a non-trivial factor of `n` by checking:
#
#     gcd(f(n) - f(k), n)
#
# for several small k >= 1, where f(n) is a C-finite sequence.

# In this method, we take f(n) to be a constant square matrix raised to power n.

# However, this method is too slow in practice for large n.

func fibonacci_matrix(k) is cached {
    Matrix.build(k, { |i,j|
        #((i == k-1) || (i == j-1)) ? (i - j + 1) : (i + j + 1)
        ((i == k-1) || (i == j-1)) ? (i - j - 1) : (i + j + 1)
        #((i == k-1) || (i == j-1)) ? (i - j) : (i + j + 1)
        #((i == k-1) || (i == j-1)) ? -1 : 1
        #((i == k-1) || (i == j-1)) ? 1 : 0
    })
}

func f(n, m, k) {
    fibonacci_matrix(k).powmod(n,m)
}

func fibonacci_matrix_factor(n, tries = 1e2) {

    say "Factoring: #{n}"

    var order = n.ilog(2)
    var z = f(n, n, order)

    tries.times { |k|

        say "Testing: #{k}"

        for v in (z - f(k, n, order) -> flat) {
            var g = gcd(v, n)
            return g if (g.is_between(2, n-1))
        }
    }

    return 1
}

say fibonacci_matrix_factor(101*503)
say fibonacci_matrix_factor(503*863)
#say fibonacci_matrix_factor(2**32 + 1)
#say fibonacci_matrix_factor(2695409723)
#say fibonacci_matrix_factor(1489390523)
#say fibonacci_matrix_factor(1e5.random_prime * 1e5.random_prime)