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An Attack on Small Private Keys of RSA Based on Euclidean Algorithm

In this paper, we describe an attack on RSA cryptosystem which is based on Euclid’s algorithm. Given a public key (n, e) with corresponding private key d such that e has the same order of magnitude as n and one of the integers k = (ed − 1)/φ(n) and e − k has at most one-quarter as many bits as e, it computes the factorization of n in deterministic time O((logn)) bit operations. MSC 2010: 94A60, 11T71, 11Y16.