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

  • Martin Dietzfelbinger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1046)

Abstract

Let u, m≥1 be arbitrary integers and let ku. The central result of this paper is that the multiset H={itha,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 xU, 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 x1, x2 are distinct elements of U and i1,i2M are arbitrary, and if h is chosen at random from H, then ¦Prob (h(x1)=i1h(x2)=i2-1/m2¦≤(1/2km)2≤1/4m2. 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.

Keywords

Hash Function Prime Number Finite Field Universal Class Integer Arithmetic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Martin Dietzfelbinger
    • 1
  1. 1.Fachbereich InformatikUniversität DortmundDortmundGermany

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