Universal hashing and k-wise independent random variables via integer arithmetic without primes

Abstract

Let u, m≥1 be arbitrary integers and let k≥u. The central result of this paper is that the multiset H={ith_a,b¦0≤a, b<km} of functions from U=0,..., u }-1 to M={0,..., m −1, where h _a,b (x)=((ax+b) mod km) div k, for x∈ U, is a (c, 2)-universal class of hash functions from U to M in the sense of Carter and Wegman [7, 25], with c=5/4. More precisely, we show that if x ₁, x ₂ are distinct elements of U and i ₁,i ₂∈ M are arbitrary, and if h is chosen at random from H, then ¦Prob (h(x ₁)=i ₁ ∧ h(x ₂)=i ₂-1/m²¦≤(1/2km)²≤1/4m ². Among the many known constructions of (c, 2)-universal classes there was none that would get by with such a small number of pure integer arithmetic operations without the assumption that a prime number of size the order of¦U¦ or at least ¦M¦ was available. — Varying this result, we obtain: (a) two-independent sequences of random variables; (b) universal hash classes of higher degree (“(c, l)-universal” classes) and l-wise independent random variables, for l ≥ 2; (c) algorithms for static and dynamic perfect hashing with an optimal number of random bits; all using pure integer arithmetic without the need for providing prime numbers (arbitrary or random) of a certain size. It should be noted that the focus here is not on minimizing the size of the probability space used, as in much of the recent work on “almost k-independent random variables”, but on the realization of such variables or hash classes using the most natural and most widely available operations, viz., integer arithmetic.