The authors consider hardware-oriented message authentication, more specifically universal hash functions. The authors propose a new type of constructions that appear promising. These constructions are based on the framework of universal hash functions, Toeplitz matrices and ε-biased sample spaces. Some new theoretical results in this area are derived. The new constructions come at the price of not being able to prove the exact substitution probability. The expected probability is examined both through theoretical methods as well as through simulation.