 
Summary: Linear Hash Functions
Noga Alon
Martin Dietzfelbinger
Peter Bro Miltersen
Erez Petrank§
G´abor Tardos¶
Abstract
Consider the set H of all linear (or affine) transformations between two vector spaces over a
finite field F. We study how good H is as a class of hash functions, namely we consider hashing
a set S of size n into a range having the same cardinality n by a randomly chosen function
from H and look at the expected size of the largest hash bucket. H is a universal class of hash
functions for any finite field, but with respect to our measure different fields behave differently.
If the finite field F has n elements then there is a bad set S F2
of size n with expected
maximal bucket size (n1/3
). If n is a perfect square then there is even a bad set with largest
bucket size always at least
n. (This is worst possible, since with respect to a universal class
of hash functions every set of size n has expected largest bucket size below
