A Unified Cryptanalysis of Generalized RSA Variants: The Monomial Explosion and Asymptotic Ineffectiveness of Prime Sharing

We present a unified lattice attack for RSA variants that satisfy the generalized keyequation eu ≡z (mod (pn−1)(qn−1)) for arbitrary integers n ≥1. Building upon the foundational framework established by Rahmani et al. (2026), we simultaneously incorporatepartial key exposure (an approximation ˜ p with |˜p−p|≤Nσ) and prime bit sharing of theform p ≡q (mod 2ρ). A central result of our work is an unconditional proof that primebit sharing does not alter the asymptotic lattice volume capacity within this generalized model: if 2ρ = Nβ, the parameter β cancels exactly from the asymptotic bound for the unraveled auxiliary variable w. By mapping the true algebraic rank over the rational field (Q), we demonstrate that higher-order shift extensions (m≥2) trigger an under-determined monomial explosion rather than a strict rank deficiency. Empirical tests confirm that our optimized, dimensionally collapsed model (m= 1) natively contains this ambient monomial space, extracting secret root parameters over a production-grade 2048-bit RSA modulus in less than a millisecond.