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Let \(N=pq\) be the product of two balanced prime numbers p and q. In 2002, Elkamchouchi, Elshenawy, and Shaban introduced an interesting RSA-like cryptosystem that, unlike the classical RSA key equation \(ed - k (p-1)(q-1) = 1\) , uses the key equation \(ed - k (p^2-1)(q^2-1) = 1\) . The scheme was further extended by Cotan and Teşeleanu to a variant that uses the key equation \(ed - k (p^n-1)(q^n-1) = 1\) , where \(n \ge 1\) . Furthermore, they provide a continued fractions attack that recovers the secret key d if \(d < N^{0.25n}\) . In this paper we improve this bound using a lattice based method. Moreover, our method also leads to the factorisation of the modulus N, while the continued fractions one does not (except for \(n=1,2,3,4\) ).

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A Lattice Attack Against a Family of RSA-Like Cryptosystems

  • 会议论文

  • 发表日期：

    2025年

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  • George Teşeleanu

摘要

Let \(N=pq\) be the product of two balanced prime numbers p and q. In 2002, Elkamchouchi, Elshenawy, and Shaban introduced an interesting RSA-like cryptosystem that, unlike the classical RSA key equation \(ed - k (p-1)(q-1) = 1\) , uses the key equation \(ed - k (p^2-1)(q^2-1) = 1\) . The scheme was further extended by Cotan and Teşeleanu to a variant that uses the key equation \(ed - k (p^n-1)(q^n-1) = 1\) , where \(n \ge 1\) . Furthermore, they provide a continued fractions attack that recovers the secret key d if \(d < N^{0.25n}\) . In this paper we improve this bound using a lattice based method. Moreover, our method also leads to the factorisation of the modulus N, while the continued fractions one does not (except for \(n=1,2,3,4\) ).