GADS-MCL Modular Congruence Constraints on Integer Factorization

We present two algebraic results that jointly reduce the effective search space in the General Number Field Sieve (GNFS) lattice sieving phase. For any integer N with residue r modulo 42: (1) every factorization N = D * k with gcd(D,42) = gcd(k,42) = 1 satisfies that the residue pair (D mod 42, k mod 42) belongs to a computable set P(r) of at most 8 pairs; (2) the identity CN - t = x(42y + b) + ay implies that (CN - t) modulo D equals ay exactly. Update Note: Extensive micro-benchmarking has confirmed these advantages within the sieving workflow. The algebraic exactness and pre-filter efficiency have been verified across multiple scales, providing a robust reduction in effective search area and confirming that the modular overhead is negligible compared to the total throughput gain.