Towards Enhanced Quantum Resistance for RSA via Constrained Rényi Entropy Optimization: A Theoretical Framework for Backward-Compatible Cryptography
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908880658759680 |
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| author | Xu, Ruopengyu Liu, Chenglian |
| author_facet | Xu, Ruopengyu Liu, Chenglian |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_00840 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Towards Enhanced Quantum Resistance for RSA via Constrained Rényi Entropy Optimization: A Theoretical Framework for Backward-Compatible Cryptography Xu, Ruopengyu Liu, Chenglian Cryptography and Security Number Theory Quantum Physics 94A60, 11Y05, 11T71, 81P94, 68Q12 E.3; F.2.1; G.3; K.6.5; C.1.m 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. |
| title | Towards Enhanced Quantum Resistance for RSA via Constrained Rényi Entropy Optimization: A Theoretical Framework for Backward-Compatible Cryptography |
| topic | Cryptography and Security Number Theory Quantum Physics 94A60, 11Y05, 11T71, 81P94, 68Q12 E.3; F.2.1; G.3; K.6.5; C.1.m |
| url | https://arxiv.org/abs/2508.00840 |