Universal Hash Families

Emin Karayel 🌐

February 20, 2022

This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.


A k-universal hash family is a probability space of functions, which have uniform distribution and form k-wise independent random variables. They can often be used in place of classic (or cryptographic) hash functions and allow the rigorous analysis of the performance of randomized algorithms and data structures that rely on hash functions. In 1981 Wegman and Carter introduced a generic construction for such families with arbitrary k using polynomials over a finite field. This entry contains a formalization of them and establishes the property of k-universality. To be useful the formalization also provides an explicit construction of finite fields using the factor ring of integers modulo a prime. Additionally, some generic results about independent families are shown that might be of independent interest.
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Theories of Universal_Hash_Families