A Note on Universal Classes of Hash Functions

In a recent paper, Carter and Wegman [2] defined universal classes of hash functions and studied their properties. In this note, we exhibit several universal classes of hash functions, and give a slightly different definition which leads to collections of hash functions which are, in at least one sense, optimal. The notation is that of [2] : A and B denote finite sets with a = IAl > lB1 = b, and H denotes a collection of functions from A and B. Throughout this paper, x and y denote distinct elements of A, and &(x, y) = 1 if f(x) = f(y), and Sf(x, y) = 0 otherwise. In [2], it is shown that for any collection H,