## Cryptology ePrint Archive: Report 2003/017

Perfect Hash Families with Few Functions

Simon R. Blackburn

Abstract: An {\em $(s;n,q,t)$-perfect hash family} is a set of functions $\phi_1,\phi_2,\ldots ,\phi_s$ from a set $V$ of cardinality $n$ to a set $F$ of cardinality $q$ with the property that every $t$-subset of $V$ is injectively mapped into $F$ by at least one of the functions $\phi_i$.

The paper shows that the maximum value $n_{s,t}(q)$ that $n$ can take for fixed $s$ and $t$ has a leading term that is linear in $q$ if and only if $t>s$. Moreover, for any $s$ and $t$ such that $t>s$, the paper shows how to calculate the coefficient of this linear leading term; this coefficient is explicitly calculated in some cases. As part of this process, new classes of good perfect hash families are constructed.

Category / Keywords: combinatorial cryptography