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**2024/1313**

**Title:**

A Lattice Attack Against a Family of RSA-like Cryptosystems

**Authors:**
George Teseleanu

**Abstract:**

Let N=pq be the product of two balanced prime numbers p and q. In 2002, Elkamchouchi, Elshenawy, and Shaban introduced an interesting RSA-like cryptosystem that, unlike the classical RSA key equation ed - k (p-1)(q-1) = 1, uses the key equation ed - k (p^2-1)(q^2-1) = 1. The scheme was further extended by Cotan and Te\c seleanu to a variant that uses the key equation ed - k (p^n-1)(q^n-1) = 1, where n \geq 1. Furthermore, they provide a continued fractions attack that recovers the secret key d if d < N^{0.25n}. In this paper we improve this bound using a lattice based method. Moreover, our method also leads to the factorisation of the modulus N, while the continued fractions one does not (except for n=1,2,3,4).

**ePrint:**
https://eprint.iacr.org/2024/1313

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