New Attacks on the RSA Cryptosystem

Abstract. This paper presents three new attacks on the RSA cryp-tosystem. The first two attacks work when k RSA public keys (Ni, ei) are such that there exist k relations of the shape eix−yiφ(Ni) = zi or of the shape eixi − yφ(Ni) = zi where Ni = piqi, φ(Ni) = (pi − 1)(qi − 1) and the parameters x, xi, y, yi, zi are suitably small in terms of the prime factors of the moduli. We show that our attacks enable us to simulta-neously factor the k RSA moduli Ni. The third attack works when the prime factors p and q of the modulus N = pq share an amount of their least significant bits (LSBs) in the presence of two decryption exponents d1 and d2 sharing an amount of their most significant bits (MSBs). The three attacks improve the bounds of some former attacks that make RSA insecure