Computing the RSA secret key is deterministic polynomial time equivalent to factoring

Abstract

We address one of the most fundamental problems concerning the RSA cryptoscheme: Does the knowledge of the RSA public key/ secret key pair (e, d) yield the factorization of N = pq in polynomial time? It is well-known that there is a probabilistic polynomial time algorithm that on input (N, e, d) outputs the factors p and q. We present the first deterministic polynomial time algorithm that factors N provided that e, d < φ (N) and that the factors p, q are of the same bit-size. Our approach is an application of Coppersmith's technique for finding small roots of bivariate integer polynomials. © International Association for Cryptologic Research 2004.