In this paper we consider very fast evaluation of strongly universal hash functions, or equivalently, authentication codes. We show how it is possible to modify some known families of hash functions into a form such that the evaluation is similar to "bucket hashing", a technique for very fast hashing introduced by Rogaway. Rogaway's bucket, hash family has a huge key size, which for common parameter choices can be more than a hundred thousand bits. The proposed hash families have a key size that is close to the key size of the theoretically best known constructions, typically a few hundred bits, and the evaluation has a time complexity that is similar to bucket hashing.