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

• Authors: Palash Sarkar
• Venue: Designs, Codes and Cryptography
• Year: 2011

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.