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.

翻译：量子计算的出现对RSA密码体系构成了严峻威胁，因为Shor算法能在多项式时间内完成整数分解。尽管后量子密码标准提供了长期解决方案，但其部署面临显著的兼容性和基础设施挑战。本文提出约束Rényi熵优化（CREO）框架，这是一种在保持完全向后兼容性的同时，可能增强RSA抗量子能力的数学方法。通过约束RSA素数间距（$|p-q| < γ\sqrt{pq}$），CREO降低了Shor算法中量子态的可区分性，该性质可通过Rényi熵进行量化。我们的分析表明：对于$k$位模数且$γ= k^{-1/2+ε}$时，可靠周期提取所需的量子测量次数按$Ω(k^{2+ε})$增长，而在理想化假设下标准RSA仅需$\mathcal{O}(k^3)$次测量。这代表了量子资源需求的系统性增长。该框架通过素数间隙定理为此类素数的构造性存在证明提供支持，并建立了与格基问题的概念性安全关联。CREO为在后量子标准长期过渡期间探索向后兼容的密码增强提供了新的研究方向，为强化广泛部署的RSA基础设施提供了一条数学基础扎实的路径，且无需立即更换协议或基础设施。