Unified Classical Security Bounds for RSA: Integrating Factorization, Structural Weakness, Leakage, and Fault Resilience

RSA security, traditionally evaluated through modulus size and asymptotic factoring complexity, most commonly using General Number Field Sieve (GNFS) estimates. However, practical deployments reveal that effective security is influenced not only by mathematical hardness but also by structural parameter choices, implementation leakage, and fault resilience. This work presents a Unified Classical Security Bound (UCSB) framework that models RSA security as a continuous, multi-layer degradation function rather than a fixed bit-level metric. We formally derive security bounds from GNFS complexity and integrate structural weaknesses (e.g., low-exponent exposure and prime imbalance), information-theoretic leakage modeling, and fault-induced compromise probability into a single minimum-bound formulation. The framework is supported by formal theorems establishing weakest-layer dominance and quantifying entropy reduction under leakage and fault scenarios. Numerical evaluations across common modulus sizes (1024–4096 bits) demonstrate that while a 2048-bit modulus provides approximately 116.9-bit resistance against classical factoring, improper parameterization or undetected CRT faults can drastically reduce effective security, in extreme cases collapsing it to near-trivial levels. The results show that RSA security is conditional and configuration-dependent rather than solely determined by key length. By integrating asymptotic complexity, structural robustness, implementation discipline, and fault detection into a unified analytical model, this work provides a rigorous and optimization-ready method for evaluating and strengthening classical RSA deployments under realistic adversarial conditions.