Almost k-wise independent sample spaces and their cryptologic applications

Abstract

An almost k-wise independent sample space is a small subset of m bit sequences in which any k bits are “almost independent”. We show that this idea has close relationships with useful cryptologic notions such as multiple authentication codes (multiple A-codes), almost strongly universal hash families and almost k-resilient functions. We use almost k-wise independent sample spaces to construct new efficient multiple A-codes such that the number of key bits grows linearly as a function of k (here k is the number of messages to be authenticated with a single key). This improves on the construction of Atici and Stinson [2], in which the number of key bits is Ω(k2). We also introduce the concept of ε-almost k-resilient functions and give a construction that has parameters superior to k-resilient functions. Finally, new bounds (necessary conditions) are derived for almost k-wise independent sample spaces, multiple A-codes and balanced ε-almost k-resilient functions.