A Generalized Attack on Some Variants of the RSA Cryptosystem

Abstract

Let \(N=pq\) be an RSA modulus with unknown factorization. The RSA cryptosystem can be attacked by using the key equation \(ed-k(p-1)(q-1)=1\). Similarly, some variants of RSA, such as RSA combined with singular elliptic curves, LUC and RSA with Gaussian primes can be attacked by using the key equation \(ed- k\left( p^2-1\right) \left( q^2-1\right) =1\). In this paper, we consider the more general equation \(eu-\left( p^2-1\right) \left( q^2-1\right) v=w\) and present a new attack that finds the prime factors p and q in the case that u, v and w satisfy some specific conditions. The attack is based on Coppersmith’s technique and improves the former attacks.