Quantum-Resistant RSA Modulus Decomposition via Adaptive Rényi Entropy Optimization

This paper establishes a rigorous theoretical foundation for enhancing RSA's quantum resistance through adaptive Rényi entropy optimization in modulus decomposition. We introduce a novel number-theoretic framework that fundamentally alters RSA's vulnerability landscape against Shor's algorithm by strategically constraining prime selection to minimize Rényi entropy $\mathscr{H}_2$. Our approach features three fundamental innovations: (1) a quantum-number theoretic security model establishing an exponential relationship between prime distribution asymmetry and quantum attack complexity, (2) an adaptive prime generation algorithm producing $\mathscr{H}_2$-optimized moduli with provable security guarantees, and (3) a security reduction proof demonstrating computational equivalence to lattice-based schemes under quantum random oracle model. Theoretical analysis proves our construction achieves $\Omega(2^{k/3})$ quantum attack complexity for $k$-bit moduli while maintaining classical security assumptions equivalent to standard RSA.