A generalized attack on RSA type cryptosystems

Abstract

Let $N=pq$ be an RSA modulus with unknown factorization. Some variants of the RSA cryptosystem, such as LUC, RSA with Gaussian primes and RSA type schemes based on singular elliptic curves use a public key $e$ and a private key $d$ satisfying an equation of the form $ed- k\left(p^2-1\right)\left(q^2-1\right)=1$. In this paper, we consider the general equation $ex-\left(p^2-1\right)\left(q^2-1\right)y=z$ and present a new attack that finds the prime factors $p$ and $q$ in the case that $x$, $y$ and $z$ satisfy a specific condition. The attack combines the continued fraction algorithm and Coppersmith\u27s technique and can be seen as a generalization of the attacks of Wiener and Blömer-May on RSA