Factorization of Square-Free Integers with High Bits Known

摘要

In this paper we propose an algorithm of factoring any integer N which has k different prime factors with the same bit-length, when (1/k+2 + ∈/k(k-1)) log N high-order bits of each prime factor are given. For a fixed e, the running time of our algorithm is heuristic polynomial in (log N). Our factoring algorithm is based on a new lattice-based algorithm of solving any k-variate polynomial equation over Z, which might be an independent interest.