A Public-Only Unraveled Lattice Diagnostic Framework for Generalized RSA Variants With Shared Prime Bits and Partial Factor Leakage

This paper presents a reproducible public-only framework for analyzing generalized Rivest– Shamir–Adleman (RSA)-type key equations of the form eu ≡ z (mod (p n − 1)(q n − 1)), where N = pq, when the adversary has partial factor information p = ˜p + x and the primes share low bits, p ≡ q (mod 2ρ ). The framework uses an unraveled polynomial model with auxiliary coordinates ti = yxi and w = y(q n − p n ), allowing the interaction between partial leakage, prime sharing, and lattice monomial growth to be studied using only public or leaked inputs. The supplementary code enforces shared prime bits during key generation, constructs the matrix without using the secret primes, retains the unraveled variables, reconstructs Lenstra–Lenstra–Lovasz output as a polynomial, and evaluates the resulting relation at the true root for validation. The theoretical contribution is deliberately scoped. We prove the public modular congruence, establish a restricted cancellation of the prime-sharing exponent in the bound for w, give monomial-count results for the implemented collapsed shift family, prove a fixed-support Howgrave– Graham obstruction for public monomial multiples of fixed powers of the n = 2 polynomial, and prove a positive affine/CRT leakage Coppersmith threshold and formalize direct leakage factoring baselines with root-count, key-equation filtering, complexity, and candidate-testing optimality bounds. The framework is intended as a reproducible diagnostic and modeling tool, not as a claimed break of production RSA.Submitted and currently under review at IEEE Access