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**2024/1263**

**Title:**

A Security Analysis of Two Classes of RSA-like Cryptosystems

**Authors:**
Paul Cotan, 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 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. Another variant of RSA, presented in 2017 by Murru and Saettone, uses the key equation ed - k (p^2+p+1)(q^2+q+1) = 1.

Despite the authorsâ€™ claims of enhanced security, both schemes remain vulnerable to adaptations of common RSA attacks. Let n be an integer. This paper proposes two families of RSA-like encryption schemes: one employs the key equation ed - k (p^n-1)(q^n-1) = 1 for n > 0, while the other uses ed - k [(p^n-1)(q^n-1)]/[(p-1)(q-1)] = 1 for n > 1. Note that we remove the conventional assumption of primes having equal bit sizes. In this scenario, we show that regardless of the choice of n, continued fraction-based attacks can still recover the secret exponent. Additionally, this work fills a gap in the literature by establishing an equivalent of Wienerâ€™s attack when the primes do not have the same bit size.

**ePrint:**
https://eprint.iacr.org/2024/1263

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