Linear hash functions

Consider the set <inline-equation> <f> <sc>H</sc> </f> </inline-equation> of all linear (or affine) transformations between two vector spaces over a finite field <italic>F</italic>. We study how good <inline-equation> <f> <sc>H</sc></f></inline-equation> is as a class of hash functions, namely we consider hashing a set <italic>S</italic> of size <italic>n</italic> into a range having the same cardinality <italic>n</italic> by a randomly chosen function from <inline-equation> <f> <sc>H</sc></f></inline-equation> and look at the expected size of the largest hash bucket. <inline-equation> <f> <sc>H</sc></f></inline-equation> 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 <italic>F</italic> has <italic>n</italic> elements, then there is a bad set <italic>S</italic> <inline-equation> <f> ⊂</f></inline-equation> <italic>F</italic><supscrpt>2</supscrpt> of size <italic>n</italic> with expected maximal bucket size <inline-equation> <f> <sc>H</sc></f></inline-equation>(<italic>n</italic><supscrpt>1/3</supscrpt>). If <italic>n</italic> is a perfect square, then there is even a bad set with largest bucket size <italic>always</italic> at least <inline-equation> <f> <rad> <rcd>n</rcd></rad></f></inline-equation>. (This is worst possible, since with respect to a universal class of hash functions every set of size <italic>n</italic> has expected largest bucket size below <inline-equation> <f> <rad> <rcd>n</rcd></rad></f></inline-equation> + 1/2.) If, however, we consider the field of two elements, then we get much better bounds. The best previously known upper bound on the expected size of the largest bucket for this class was <italic>O</italic>(2<supscrpt><inline-equation> <f> <rad> <rcd>log n</rcd></rad></f></inline-equation></supscrpt>). We reduce this upper bound to <italic>O</italic>(log <italic>n</italic> log log<italic>n</italic>). Note that this is not far from the guarantee for a random function. There, the average largest bucket would be &THgr;(log <italic>n</italic>/ log log <italic>n</italic>). In the course of our proof we develop a tool which may be of independent interest. Suppose we have a subset <italic>S</italic> of a vector space <italic>D</italic> over <bold>Z</bold><subscrpt>2</subscrpt>, and consider a random linear mapping of <italic>D</italic> to a smaller vector space <italic>R</italic>. If the cardinality of <italic>S</italic> is larger than <italic>c</italic><subscrpt>ε</subscrpt>|<italic>R</italic>|log|<italic>R</italic>|, then with probability 1 - ε, the image of <italic>S</italic> will cover all elements in the range.