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**2017/1076**

**Title:**

A generalized attack on RSA type cryptosystems

**Authors:**
Martin Bunder, Abderrahmane Nitaj, Willy Susilo, Joseph Tonien

**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’s technique and can be seen as a generalization of the attacks of Wiener and Blömer-May on RSA.

**ePrint:**
https://eprint.iacr.org/2017/1076

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