Paper 2026/519

A Generalized Partial Exposure Lattice Attack Against an RSA variant Based on Cubic Pell Curves

Abstract

Nitaj and Seck recently published an RSA variant (MJAGA 2024) based on the cubic Pell equation $\mathcal{P}_c(N): u^3+cv^3+c^2w^3-3cuvw= 1$ over $\mathbb{Z}/N\mathbb{Z}$ when $N=p^rq^s$. In their cryptosystem, the public exponent $e$ and the private exponent $d$ are related to the key equation $d\equiv e^{-1}\pmod{p^{2(r-1)}q^{2(s-1)}(p-1)^2(q-1)^2}$. In AfricaCrypt 2025, Rahmani and Nitaj published a lattice attack on their scheme in the particular case of $r=s=1$ by exploiting the key equation $ed - (p-1)^2(q-1)^2 k = 1$. In this paper, we present a new generalized partial exposure lattice attack on the scheme of Nitaj and Seck by examining the key equation $eu_0 - (p-1)^2(q-1)^2 v_0 = w_0$ when some bits of $p$ or $q$ are known.