Efficient Strongly Universal and Optimally Universal Hashing

Abstract

New hash families are analyzed, mainly consisting of the hash functions h _a,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 several 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 ℤ_m.