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**2023/1356**

**Title:**

Small Private Key Attack Against 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. Elkamchouchi, Elshenawy and Shaban presented in 2002 an interesting RSA-like cryptosystem that uses the key equation ed - k (p^2-1)(q^2-1) = 1, instead of the classical RSA key equation ed - k (p-1)(q-1) = 1. The authors claimed that their scheme is more secure than RSA. Unfortunately, the common attacks developed against RSA can be adapted for Elkamchouchi \emph{et al.}'s scheme. 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) = 1, where n>0 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/2023/1356

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