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Tiny families of functions with random properties: A quality-size trade-off for hashing. (English) Zbl 0891.60010
Summary: We present three explicit constructions of hash functions, which exhibit a trade-off between the size of the family (and hence the number of random bits needed to generate a member of the family), and the quality (or error parameter) of the pseudorandom property it achieves. Unlike previous constructions, most notably universal hashing, the size of our families is essentially independent of the size of the domain on which the functions operate. The first construction is for the mixing property – mapping a proportional part of any subset of the domain to any other subset. The other two are for the extraction property – mapping any subset of the domain almost uniformly into a range smaller than it. The second and third constructions handle, respectively, the extreme situations when the range is very large or very small. We provide lower bounds showing that our constructions are nearly optimal, and mention some applications of the new constructions.

MSC:
60C05 Combinatorial probability
68P10 Searching and sorting
94C10 Switching theory, application of Boolean algebra; Boolean functions (MSC2010)
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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