New hash functions for message authentication

TLDR: It is shown that Toeplitz matrices generated by sequences drawn from small biased distributions provide hashing schemes applicable to secure message authentication and the interest of these schemes extends beyond the proposed cryptographic applications to other uses of universal hashing.

Abstract: We show that Toeplitz matrices generated by sequences drawn from small biased distributions provide hashing schemes applicable to secure message authentication. This work extends our previous results from Crypto'94 [4] where an authentication scheme based on Toeplitz matrices generated by linear feedback shift registers was presented. Our new results have as special case the LFSR-based construction but extend to a much wider and general family of sequences, including several simple and efficient constructions with close to optimal security. Examples of the new constructions include Toeplitz matrices generated by the Legendre symbols of consecutive integers modulo a prime (of size significantly shorter than required by public-key modular arithmetic) as well as other algebraic constructions. The interest of these schemes extends beyond the proposed cryptographic applications to other uses of universal hashing (including other cryptographic applications).