Hide and Seek - a Naive Factoring

I present a factoring algorithm that factors N = U V , where U < V , provably in O(N 1/3+ǫ) time. I also discuss the potential for improving this to a sub-exponential algorithm. Along the way, I consider the distribution of solutions (x, y) to xy = N mod a. 1. Algorithm-hide and seek Let N be a positive integer that we wish to factor. Say N = U V where U and V are positive integers, not necessarily prime, with 1 < U < V. For simplicity, that V < 2U so that V < (2N) 1/2 The general case, without this restriction, will be handled at the end of this section. The idea behind the algorithm is to perform trial division of N by a couple of integers, and to use information about the remainder to determine the factors U and V. Let a be a positive integer, 1 < a < N. By the division algorithm, write U = u 1 a + u 0 , with 0 ≤ u 0 < a V = v 1 a + v 0 , with 0 ≤ v 0 < a. (1.1) Assume that u 0 is relatively prime to a, and likewise for v 0 , since otherwise one easily extracts a factor of N by taking gcd(a, N). If, for a given a, we can determine u 0 , u 1 , v 0 , v 1 then we have found U and V. Consider N = u 0 v 0 mod a. One cannot simply determine u 0 and v 0 from the value of N mod a since φ(a) pairs of integers (x, y) mod a satisfy xy = N mod a (if x = mu 0 mod a, then y = m −1 v 0 mod a, where gcd(m, a) = 1). However, say a is large, a = ⌈(2N) 1/3 ⌉ > V 2/3 so that v 1 and u 1 are comparatively small, u 1 , v 1 ≤ V 1/3 , i.e. both are < a 1/2. If we consider N mod a − δ (1.2) N = U V = (u 1 δ + u 0)(v 1 δ + v 0) mod a − δ, for δ = 0, 1, we get, as solutions (x, y) to xy = N mod a − δ, two nearby points, (u 0 …