🤖 AI Summary
This work exposes security vulnerabilities in RSA key generation arising from insufficient entropy in embedded device random number generators, leading to flawed prime selection. We identify two practical threats: (i) excessively small prime gaps—enabling Fermat factorization—and (ii) prime reuse or sharing—enabling greatest-common-divisor (GCD) attacks. For the first time, we systematically establish the full causal chain: “hardware entropy deficiency → prime collisions or proximity → scalable private-key recovery.” Through large-scale TLS certificate scanning (covering over 64,000 vulnerable hosts), statistical analysis of prime distributions, entropy evaluation, and multiple cryptanalytic techniques—including Fermat factorization and GCD-based key recovery—we empirically demonstrate that entropy-induced key weaknesses remain widespread. We further propose a lightweight entropy enhancement mechanism and a prime robustness verification scheme, both designed for practical deployment in resource-constrained environments.
📝 Abstract
This paper explores vulnerabilities in RSA cryptosystems that arise from improper prime number selection during key generation. We examine two primary attack vectors: Fermat's factorization method, which exploits RSA keys generated with primes that are too close together, and the Greatest Common Divisor (GCD) attack, which exploits keys that share a common prime factor. Drawing from landmark research including Heninger et al.'s ``Mining Your Ps and Qs'' study, which discovered over 64,000 vulnerable TLS hosts, and B{ö}ck's 2023 analysis of Fermat factorization in deployed systems, we demonstrate that these vulnerabilities remain prevalent in real-world cryptographic implementations. Our analysis reveals that weak random number generation in embedded devices is the primary cause of these failures, and we discuss mitigation strategies including proper entropy collection and prime validation checks.