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

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