New hash families are analyzed, mainly consisting of the hash functions ha,b : {0,..., u - 1} → {0,..., r - 1}, x → ((ax + b) mod(kr)) div k. Universal classes of such functions have already been investigated in [5, 6], and used in severail applications, e.g. [3,9]. The new constructions which are introduced here, improve in several ways upon the former results. Some of them achieve a smaller universality parameter, i.e., two keys collide under a randomly chosen function with a smaller probability. In fact, an optimally universal hash class is presented, which means that the universality parameter achieves the minimum possible value. Furthermore, the bound of the universality parameter of a known, almost strongly universal hash family is improved, and it is shown how to reduce the size of a known class, retaining its properties. Finally, a new composition technique for constructing hash classes for longer keys is presented. Its application leads to efficient hash families which consist of linear functions over the ring of polynomials over Zm.