Towards Enhanced Quantum Resistance for RSA via Constrained Rényi Entropy Optimization: A Theoretical Framework for Backward-Compatible Cryptography
It addresses the problem of quantum vulnerability for widely deployed RSA infrastructure, offering a backward-compatible enhancement during the transition to post-quantum standards, though it is incremental as it builds on existing RSA with theoretical modifications.
This paper tackles the threat of quantum computing to RSA cryptography by proposing the Constrained Rényi Entropy Optimization (CREO) framework, which constrains RSA primes to reduce quantum state distinguishability in Shor's algorithm, resulting in an increase in required quantum measurements from O(k^3) to Ω(k^{2+ε}) for a k-bit modulus.
The advent of quantum computing poses a critical threat to RSA cryptography, as Shor's algorithm can factor integers in polynomial time. While post-quantum cryptography standards offer long-term solutions, their deployment faces significant compatibility and infrastructure challenges. This paper proposes the Constrained Rényi Entropy Optimization (CREO) framework, a mathematical approach to potentially enhance the quantum resistance of RSA while maintaining full backward compatibility. By constraining the proximity of RSA primes ($|p-q| < γ\sqrt{pq}$), CREO reduces the distinguishability of quantum states in Shor's algorithm, as quantified by Rényi entropy. Our analysis demonstrates that for a $k$-bit modulus with $γ= k^{-1/2+ε}$, the number of quantum measurements required for reliable period extraction scales as $Ω(k^{2+ε})$, compared to $\mathcal{O}(k^3)$ for standard RSA under idealized assumptions. This represents a systematic increase in quantum resource requirements. The framework is supported by constructive existence proofs for such primes using prime gap theorems and establishes conceptual security connections to lattice-based problems. CREO provides a new research direction for exploring backward-compatible cryptographic enhancements during the extended transition to post-quantum standards, offering a mathematically grounded pathway to harden widely deployed RSA infrastructure without requiring immediate protocol or infrastructure replacement.