Deterministic Polynomial-Time Equivalence of Computing the RSA Secret Key and Factoring

Abstract. We address one of the most fundamental problems concerning the RSA cryptosystem: does the knowledge of the RSA public and 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). Our approach is an application of Coppersmith’s technique for finding small roots of univariate modular polynomials.

• Type: Journal
• Year: 2007
• Where: JOC
• Authors: Jean-Sébastien Coron, Alexander May
• Keywords: JOC 2007, Polynomial Time, Probabilistic Polynomial Time, RSA Cryptosystem