Introduction

Recent years have seen numerous interesting applications of combinatorics to cryptography. In particular, combinatorial designs have played an important role in the study of such topics in cryptography as secrecy and authentication codes, secret sharing schemes, and resilient functions. The purpose of this paper is to elucidate some of these connections. This is not intended to be an exhaustive survey, but rather a sampling of some research topics in which I have a personal interest.

Some other related surveys include applications of error-correcting codes to cryptography (Sloane [60]), applications of finite geometry to crpytography (Beutelspacher [6]) and applications of combinatorial designs to computer science (Colbourn and Van Oorschot [16]).

In this introduction we will define the relevant concepts from design theory that we will have occasion to use. Beth, Jungnickel and Lenz [4] is a good general reference on design theory.

Let 1 ≤ t ≤ k < v. An Sλ(t, k, v) is defined to be a pair (X, β), where X is a v-set (of points) and β is a collection of κ-subsets of X (called blocks), such that every t-subset of points occurs in exactly λ blocks. An Sλ(t, k, v) is called simple if no block occurs more than once in β.