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**2022/1241**

**Title:**

Continued Fractions Applied to a Family of RSA-like Cryptosystems

**Authors:**
George Teseleanu, Paul Cotan

**Abstract:**

Let N=pq be the product of two balanced prime numbers p and q. Murru and Saettone presented in 2017 an interesting RSA-like cryptosystem that uses the key equation ed - k (p^2+p+1)(q^2+q+1) = 1, instead of the classical RSA key equation ed - k (p-1)(q-1) = 1. The authors claimed that their scheme is immune to Wiener’s continued fraction attack. Unfortunately, Nitaj \emph{et. al.} developed exactly such an attack. In this paper, we introduce a family of RSA-like encryption schemes that uses the key equation ed - k [(p^n-1)(q^n-1)]/[(p-1)(q-1)] = 1, where n>1 is an integer. Then, we show that regardless of the choice of n, there exists an attack based on continued fractions that recovers the secret exponent.

**ePrint:**
https://eprint.iacr.org/2022/1241

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