Secure RSA Key Generation Using a 1D Hybrid Logistic-Tent Map and SHA-256 Hash Conditioning

Abstract

The RSA cryptosystem remains a cornerstone of modern public-key cryptography; however, its security fundamentally depends on the quality of prime generation and the unpredictability of the underlying random number source. Conventional Pseudorandom Number Generators (PRNGs) rely on deterministic algorithms with limited entropy sources, potentially introducing patterns exploitable through cryptanalytic attacks. This study proposes an RSA key generation framework that employs a Hybrid Logistic-Tent Map (LTM) with SHA-256 hash conditioning. The Hybrid LTM integrates the dynamic properties of both logistic and tent chaotic maps through modular composition, achieving a Lyapunov exponent of 1.9524 compared to 0.9932 for the logistic map alone. The multi-bit extraction scheme converts chaotic trajectories into binary sequences, and SHA-256 conditioning eliminates statistical bias inherent in finite-precision chaotic implementations. Experimental results across 512-bit and 1024-bit key sizes demonstrate that SHA-256 conditioning improves the Hybrid LTM from 2/7 to 7/7 NIST SP 800-22 test pass rate while maintaining Shannon entropy of 7.9869 bits/byte. All 60 generated key pairs satisfy safety conditions for Wiener, Fermat, and GCD-based attacks, with a key sensitivity Hamming distance ratio of 50.4% under 10-15 initial condition perturbation.