A trade-off between collision probability and key size in universal hashing using polynomials

Abstract

Let ${\mathbb{F}}$ be a finite field and suppose that a single element of ${\mathbb{F}}$ is used as an authenticator (or tag). Further, suppose that any message consists of at most L elements of ${\mathbb{F}}$. For this setting, usual polynomial based universal hashing achieves a collision bound of ${(L-1)/|\mathbb{F}|}$ using a single element of ${\mathbb{F}}$ as the key. The well-known multi-linear hashing achieves a collision bound of ${1/|\mathbb{F}|}$ using L elements of ${\mathbb{F}}$ as the key. In this work, we present a new universal hash function which achieves a collision bound of ${m\lceil\log_m L\rceil/|\mathbb{F}|, m\geq 2}$, using ${1+\lceil\log_m L\rceil}$ elements of ${\mathbb{F}}$ as the key. This provides a new trade-off between key size and collision probability for universal hash functions.