Paper 2026/1357

A refined cryptanalytic attack against a generalized cubic Pell RSA scheme

Abstract

In 2022, Cotan and Te{\c{s}}eleanu proposed an alternative RSA construction in which the modulus takes the form $N = pq$ and the exponents $(e,d)$ are linked through $ ed -1\equiv 0 \pmod{\psi_n(N)},\ n \geq 2, $ where $ \psi_n(N) = \frac{(p^n-1)(q^n-1)}{(p-1)(q-1)}. $ Their scheme was subsequently examined by Nitaj et al. at Africacrypt 2024, who demonstrated that the system becomes vulnerable when the secret exponent $d$ is less than $N^{0.292(n-1)}$. In the present work, we extend this line of cryptanalysis by developing a lattice-based technique targeting the same key congruence. Our approach succeeds in recovering the private parameters even for secret exponents greater than $N^{0.292(n-1)}$, thereby surpassing the previously known boundary. The factorization of $N$ is achieved in polynomial time, provided that its divisors $p$ and $q$ share a suitably portion of their least significant bits.