Universal Hash Functions & Hard Core Bits

In this paper we consider the bit-security of two types of universal hash functions: linear functions on GFF2 n ] and linear functions on the integers modulo a prime. We show individual security for all bits in the rst case and for the O(log n) least signiicant bits in the second case. Both types of functions are shown to have O(log n) simultaneous secure bits. For the second type of functions, primes of length (n) are needed. Together with the Goldreich-Levin theorem, this shows that all the common types of universal hash functions provide so called hard-core bits.