Towards Enhanced Quantum Resistance for RSA via Constrained R\'enyi Entropy Optimization: A Theoretical Framework for Backward-Compatible Cryptography

TL;DR

This paper introduces the CREO framework, a mathematical approach to modify RSA primes to increase quantum resistance while maintaining backward compatibility, thereby raising the quantum resource requirements for attacking RSA.

Contribution

The paper proposes the CREO framework, constraining RSA primes to enhance quantum resistance without changing existing infrastructure, supported by theoretical proofs and security analysis.

Findings

Quantum measurement complexity for CREO-modified RSA scales as Ω(k^{2+ε})

CREO primes can be constructed using prime gap theorems

Framework offers a systematic way to enhance RSA's quantum resistance

Abstract

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 R\'enyi 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 ($|p-q| < \gamma \sqrt{pq}$), CREO reduces the distinguishability of quantum states in Shor's algorithm, as quantified by R\'enyi entropy. Our analysis demonstrates that for a $k$-bit modulus with $\gamma = k^{-1/2+\epsilon}$, the number of quantum measurements required for reliable period extraction scales as $\Omega(k^{2+\epsilon})$, 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.