Stronger security bounds for wegman-carter-shoup authenticators

• Authors: Daniel J. Bernstein
• Venue: EUROCRYPT'05 Proceedings of the 24th annual international conference on Theory and Applications of Cryptographic Techniques
• Year: 2005

Shoup proved that various message-authentication codes of the form (n,m) ↦ h(m) + f(n) are secure against all attacks that see at most $\sqrt{1/\epsilon}$ authenticated messages. Here m is a message; n is a nonce chosen from a public group G; f is a secret uniform random permutation of G; h is a secret random function; and ε is a differential probability associated with h. Shoup's result implies that if AES is secure then various state-of-the-art message-authentication codes of the form (n,m) ↦h(m)+AESk(n) are secure up to $\sqrt{ 1/\epsilon}$ authenticated messages. Unfortunately, $\sqrt{ 1/\epsilon}$ is only about 250 for some state-of-the-art systems, so Shoup's result provides no guarantees for long-term keys. This paper proves that security of the same systems is retained up to $\sqrt{\#G}$ authenticated messages. In a typical state-of-the-art system, $\sqrt{\#G}$ is 264. The heart of the paper is a very general “one-sided” security theorem: (n,m) ↦ h(m) + f(n) is secure if there are small upper bounds on differential probabilities for h and on interpolation probabilities for f.