The RSA (Rivest-Shamir-Adleman) cryptosystem is a fundamental algorithm of public key cryptography and is widely used across various information domains. For an RSA modulus represented as \(N = pq\) , with its factorization remaining unknown, security vulnerabilities arise when attackers exploit the key equation \(ed-k(p-1)(q-1)=1\) . To enhance the security, Murru and Saettone introduced cubic Pell RSA—a variant of RSA based on the cubic Pell equation, where the key equation becomes \(ed-k(p^2+p+1)(q^2+q+1)=1\) . In this paper, we further investigate the security implications surrounding the generalized key equation \(eu-(p^2+p+1)(q^2+q+1)v=w\) . We present a novel attack strategy aimed at recovering the prime factors p and q under specific conditions satisfied by u, v, and w. Our generalized attack employs lattice-based Coppersmith’s techniques and extends several previous attack scenarios, thus deepening the understanding of mathematical cryptanalysis.