Abstract
Let \(N=pq\) be an RSA modulus with balanced prime factors. In 2018, Murru and Saettone presented a variant of the RSA cryptosystem based on a cubic Pell equation in which the public key (N, e) and the private key (N, d) satisfy \(ed\equiv 1\pmod {\left( p^2+p+1\right) \left( q^2+q+1\right) }\). They claimed that the classical small private attacks on RSA such as Wiener’s continued fraction attack do not apply to their scheme. In this paper, we show that, on the contrary, Wiener’s method as well as the small inverse problem technique of Boneh and Durfee can be applied to attack their scheme. More precisely, we show that the proposed variant of RSA can be broken if \(d<N^{0.5694}\). This shows that their scheme is in reality more vulnerable than RSA, where the bound of vulnerability is \(d<N^{0.292}\).
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Nitaj, A., Ariffin, M.R.B.K., Adenan, N.N.H., Abu, N.A. (2021). Classical Attacks on a Variant of the RSA Cryptosystem. In: Longa, P., Ràfols, C. (eds) Progress in Cryptology – LATINCRYPT 2021. LATINCRYPT 2021. Lecture Notes in Computer Science(), vol 12912. Springer, Cham. https://doi.org/10.1007/978-3-030-88238-9_8
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