Quantum-Resistant RSA Modulus Decomposition via Adaptive Rényi Entropy Optimization

• URL: http://arxiv.org/abs/2508.00840v2
• Date: Tue, 05 Aug 2025 02:07:27 GMT
• Title: Quantum-Resistant RSA Modulus Decomposition via Adaptive Rényi Entropy Optimization
• Abstract summary: This paper explores a theoretical approach to enhance RSA's resistance against quantum attacks.<n>We develop a framework where primes are generated with controlled proximity.<n>Main contributions include establishing a connection between prime distribution properties and quantum attack complexity.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: This paper explores a theoretical approach to enhance RSA's resistance against quantum attacks by optimizing prime selection through R\'enyi entropy constraints. We develop a framework where primes are generated with controlled proximity ($|p-q| < \gamma\sqrt{pq}$) to minimize the collision entropy $\mathscr{H}_2$ of the quantum period-finding operator. The main contributions include: (1) establishing a connection between prime distribution properties and quantum attack complexity via Maynard's prime gap theorem, (2) providing a constructive proof for prime existence under entropy constraints, and (3) demonstrating security reduction to ideal lattice problems under the quantum random oracle model. Theoretical analysis suggests that for $k$-bit moduli with $\gamma < k^{-1/2+\epsilon}$, Shor's algorithm requires $\Omega(\gamma^{-1}k^{3/2})$ quantum operations while maintaining classical security equivalent to standard RSA. Key Enhancements: (1) Prime existence proof via Maynard's theorem (Theorem 3.1), (2) Ideal lattice embedding for SVP reduction (Theorem 5.3), (3) Quantum Fano bound for information-theoretic analysis (Theorem 6.3), (4) Multi-prime RSA extension (Section 7.3).

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