A Generalized Attack on Some Variants of the RSA Cryptosystem

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(p2−1)(q2−1)=1 . In this paper, we consider the more general equation eu−(p2−1)(q2−1)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\u27s technique and improves the former attacks