/
congruence_of_powers_factorization_method.sf
131 lines (101 loc) · 4.13 KB
/
congruence_of_powers_factorization_method.sf
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#!/usr/bin/ruby
# Daniel "Trizen" Șuteu
# Date: 05 July 2019
# https://github.com/trizen
# A simple factorization method, based on congruences of powers.
# Given a composite integer `n`, if we find:
#
# a^k == b^k (mod n)
#
# for some k >= 2, then gcd(a-b, n) may be a non-trivial factor of n.
# See also:
# https://trizenx.blogspot.com/2019/08/special-purpose-factorization-algorithms.html
define MIN_FACTOR = 1e6 # ignore small factors
func cgpow_factor (n) {
var params = []
var orig = n
var range = (2 .. min(n.ilog2, 15) -> flip)
func process_congruence(root, e) {
for j in (-1, 0, 1) {
var k = (root + j)
var u = powmod(k, e, n)
for z in (u, n-u) {
if (is_power(z)) {
var t = perfect_power(z)
var r1 = z.iroot(t)
var r2 = z.iroot(e)
params << [r1, t, k, e]
params << [r2, e, k, e]
}
}
}
}
range.each {|e|
var root = n.iroot(e)
process_congruence(root, e)
}
range.each {|root|
var e = n.ilog(root)
process_congruence(root, e)
}
var f = func (r, e1, k, e2) {
var factors = Set()
var d1 = e1.divisors.map {|d| r**d }
var d2 = e2.divisors.map {|d| k**d }
for x in (d1), y in (d2) {
for j in (-1, 1) {
var t = (x - j*y)
var g = gcd(t, n)
if (g>MIN_FACTOR && g<n) {
factors << g
}
}
}
factors.to_a
}
var factors = []
var divisors = params.uniq.map { f(_...) }.flat.sort.uniq
divisors.each {|d|
var g = gcd(n, d)
if (g>MIN_FACTOR && g<n) {
while (n%g == 0) {
n /= g
factors << g
}
}
}
factors << (orig / factors.prod)
factors.sort
}
if (ARGV) {
say cgpow_factor(Num(ARGV[0]))
return 1
}
say cgpow_factor(ipow(2, 256) - 1)
say cgpow_factor(ipow(10, 120) + 1)
say cgpow_factor(ipow(10, 120) - 1)
say cgpow_factor(ipow(10, 120) - 25)
say cgpow_factor(ipow(10, 105) - 1)
say cgpow_factor(ipow(10, 105) + 1)
say cgpow_factor(ipow(10, 120) - ipow(2134, 2))
say cgpow_factor((ipow(2, 128) - 1) * (ipow(2, 256) - 1))
say cgpow_factor(ipow(ipow(4, 64) - 1, 3) - 1)
say cgpow_factor((2**128 - 1) * (3**128 - 1))
say cgpow_factor((5**48 + 1) * (3**120 + 1))
say cgpow_factor((5**48 + 1) * (3**120 - 1))
say cgpow_factor((5**48 - 1) * (3**120 + 1))
say cgpow_factor((45**120 + 35**420))
__END__
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[100000001, 9999999900000001, 99999999000000009999999900000001, 10000000099999999999999989999999899999999000000000000000100000001]
[1000001, 1202981, 1587789, 2906161, 99009901, 1000000000001, 10989010989011, 165573604901641, 9999000099990001, 100009999999899989999000000010001]
[999999999999999999999999999999999999999999999999999999999995, 1000000000000000000000000000000000000000000000000000000000005]
[1111111, 8392599, 119152601, 900900990991, 900009090090909909099991, 1109988789001111109989898989900111110998878900111]
[1478477, 4734601891, 156985855573, 999990000099999000009999900001, 910009191000909089989898989899909091000919100091]
[999999999999999999999999999999999999999999999999999999997866, 1000000000000000000000000000000000000000000000000000000002134]
[4294836225, 4294967297, 4294967297, 4295098369, 18446744073709551617, 18446744073709551617, 340282366920938463463374607431768211457]
[340282366920938463463374607431768211454, 115792089237316195423570985008687907852929702298719625575994209400481361428481]
[41, 1048560, 1979408, 5570645, 6700417, 43046722, 926510094425921, 18446744073709551617, 1716841910146256242328924544641]
[1273028, 43040161, 76293945313, 1852737802355521, 240031591394168814433, 3434207167913380326433387290241]
[1013824, 1801025, 2828617, 16068403, 47763361, 1743392201, 76293945313, 50446744628921761, 240031591394168814433]
[1889856, 3390842, 43040161, 61132969, 1852737802355521, 59509277587890001, 3434207167913380326433387290241]