Refined analysis and improvements on some factoring algorithms

Abstract

By combining the principles of known factoring algorithms we obtain some improved algorithms which by heuristic arguments all have a time bound O(exp √c ln n ln ln n) for various constants c≥3. In particular, Miller's method of solving index equations and Shanks's method of computing ambiguous quadratic forms with determinant −n can be modified in this way. We show how to speed up the factorization of n by using preprocessed lists of those numbers in [−u,u] and [n−u,n+u],O<<u