CSE 599 b : Cryptography ( Winter 2006 ) Lecture 14 : Cryptographic Hash Functions 17 February 2006

and thus being a universal hash function family is equivalent to having a probability distribution on functions from D to R that maps elements of D in a uniform pairwise independent fashion. Typically we will consider D = {0, 1} and R = {0, 1} form < n. The following construction due to Dietzfelbinger is particularly convenient: The space of keys is all strings K = (a, b) where a, b ∈ {0, 1} and HK(x) consists of the middlem bits of ax+ b. (ax+ b will naturally have 2n+m bits.) In keeping with our choice of considering PPT adversaries for our formal definitions we will use infinite hash function families and allow probability slop that is negligible. We will also want our hash function families to be efficiently computable. Before we consider the cryptographic versions we state a relaxation of the universal hash function family definition.