Universal hash functions are important building blocks for unconditionally secure message authentication codes. In this paper, we present a new construction of a class of e-Almost Strongly Universal2 hash functions with much smaller description (or key) length than the Wegman-Carter construction. Unlike some other constructions, our new construction has a very short key length and a security parameter e that is independent of the message length, which makes it suitable for authentication in practical applications such as Quantum Cryptography.