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Provably fast integer factoring with quasi-uniform small quadratic residues

Finding small quadratic residues modulo n, when n is a large composite number of unknown factorisation is almost certainly a computationally hard problem. This problem arises in a natural way when factoring n by the use of congruences of squares. We construct here a polynomial-time algorithm based on the use of lattices, which finds in a near uniform way quadratic residues mod n that are smaller than O(n2/3). In this way, we derive a class of integer factorisation algorithms, the fastest of which provides the best rigorously established probabilistic complexity bound for integer factorisation algorithms.