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 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.
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Cotan, P., Teşeleanu, G. (2022). Continued Fractions Applied to a Family of RSA-like Cryptosystems. In: Su, C., Gritzalis, D., Piuri, V. (eds) Information Security Practice and Experience. ISPEC 2022. Lecture Notes in Computer Science, vol 13620. Springer, Cham. https://doi.org/10.1007/978-3-031-21280-2_33
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