RSA Security

Number Theory · Axiom Academy

Understanding why RSA encryption is secure and what vulnerabilities exist RSA security depends on the integer factorization problem : given a large number n = pq , finding the prime factors p and q is computationally infeasible for sufficiently large values. The animation below shows how easy it is to multiply primes (forward), but how exponentially harder it becomes to factor the result (backward). 2. Key Sizes and Security Levels The security of RSA scales with key size. As computing power increases, we need larger keys to maintain security. The relationship between key size and security is subexponential due to the complexity of the best factoring algorithms. This animation illustrates how different key sizes provide varying levels of security against factoring attacks. 1024-bit: Deprecated - vulnerable to well-funded attackers 2048-bit: Current standard for most applications 3072-bit: Recommended for long-term security (10+ years) 4096-bit: Maximum security for highly sensitive data While direct factoring is hard, RSA can be vulnerable to attacks that exploit implementation weaknesses or poor parameter choices. Understanding these attacks is crucial for secure implementation. Small Public Exponent (e=3): Allows direct root extraction for small messages Common Modulus Attack: Using same n for multiple users enables message recovery Timing Attacks: Measuring decryption time reveals private key bits Weak Random Number Generation: Predictable primes break security entirely

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