Compare cryptographic key lengths across algorithms for equivalent security strength.
Different cryptographic algorithms achieve the same security strength with vastly different key sizes. RSA requires much longer keys than AES for equivalent security because its strength relies on the difficulty of factoring large numbers (sub-exponential attack) rather than brute-force search (exponential). ECC sits in between, offering compact keys with strong security.
NIST equivalence (approximate)
AES-128 ~ RSA-3072 ~ ECC-256 (128-bit security level)
RSA's security depends on the difficulty of integer factoring, which has sub-exponential (not exponential) attack complexity. The Number Field Sieve algorithm can factor RSA keys much faster than brute force, so RSA needs disproportionately larger keys to maintain equivalent security against the best known attacks.
In terms of brute-force keyspace, AES-256 has 2^128 times more keys than AES-128 (not twice as many). Both are considered unbreakable by brute force with current or foreseeable technology, but AES-256 provides a larger margin against potential future cryptanalytic advances or quantum computing threats.