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In 2024, based on the cubic Pell curve, Nitaj and Seck proposed a variant of the RSA cryptosystem where the modulus is in the form \(N=p^rq^s\) , and the public key e and private key d satisfy the equation \(ed\equiv 1\pmod {(p-1)^2(q-1)^2}\) . They showed that N can be factored when d is less than a certain bound that depends on r and s in the situation \(rs\ge 2\) , which is not extendable to \(r=s=1\) . In this paper, we propose a cryptanalysis of this scheme in the situation \(r=s=1\) , and give an explicit bound for d that makes the scheme insecure. The method is based on Coppersmith’s method and lattice reduction.

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Improved Cryptanalysis of an RSA Variant Based on Cubic Pell Curve

  • 会议论文

  • 发表日期：

    2026年

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  • Mohammed Rahmani,
  • Abderrahmane Nitaj

摘要

In 2024, based on the cubic Pell curve, Nitaj and Seck proposed a variant of the RSA cryptosystem where the modulus is in the form \(N=p^rq^s\) , and the public key e and private key d satisfy the equation \(ed\equiv 1\pmod {(p-1)^2(q-1)^2}\) . They showed that N can be factored when d is less than a certain bound that depends on r and s in the situation \(rs\ge 2\) , which is not extendable to \(r=s=1\) . In this paper, we propose a cryptanalysis of this scheme in the situation \(r=s=1\) , and give an explicit bound for d that makes the scheme insecure. The method is based on Coppersmith’s method and lattice reduction.