Generation of elements with small modular squares and provably fast integer factoring algorithms

Abstract:

Finding small modular squares, when the modulus is a large composite number of unknown factorization, is almost certainly a computationally hard problem. This problem arises in a natural way when factoring the modulus by the use of congruences of squares. We study here, with the help of lattices, the set of elements whose squares $\bmod n$ are small enough, less than $O({n^{2/3}})$. We obtain a precise description of the gaps between such elements, and we develop two polynomial-time algorithms that find elements with small modular squares. The first is a randomized algorithm that generates such elements in a near uniform way. We use it to derive a class of integer factorization algorithms, the fastest of which provides the best rigorously established probabilistic complexity bound for integer factorization algorithms. The second algorithm is deterministic and often finds, amongst the neighbors of a given point, the nearest one that has a small modular square.