New bounds for almost universal hash functions
Long H. Nguyen and Andrew W. Roscoe
Abstract
Using combinatorial analysis and the pigeon-hole principle, we introduce a new lower (or combinatorial) bound for the key length in an almost (both pairwise and l-wise) universal hash function. The advantage of this bound is that the key length only grows in proportion to the logarithm of the message length as the collision probability moves away from its theoretical minimum by an arbitrarily small and positive value. Conventionally bounds for various families of universal hash functions have been derived by using the equivalence between pairwise universal hash functions and error-correcting codes, and indeed in this paper another (very similar) bound for almost universal hash functions can be calculated from the Singleton bound in coding theory. However, the latter, perhaps unexpectedly, will be shown to be not as tight as our combinatorial one. To the best of our knowledge, this is the first time that combinatorial analysis has been demonstrated to yield a better universal hash function bound than the use of the equivalence, and we will explain why there is such a mismatch. This work therefore potentially opens the way to re-examining many bounds for a number of families of universal hash functions, which have been solely derived from bounds in coding theory and/or other combinatorial objects.
Details
| Journal |
To be submitted |
| Year |
2010 |
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