A generalized attack on RSA type cryptosystems

Abstract Let N = p q 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 e d − k ( p 2 − 1 ) ( q 2 − 1 ) = 1 . In this paper, we consider the general equation e x − ( p 2 − 1 ) ( q 2 − 1 ) 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 Blomer–May on RSA.

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