Combinatorial designs, sequences and cryptography

A major part of this thesis is about sequences with low, zero or constant autocorrelation function and their implications to combinatorial designs and cryptography. We discuss sequences with zero (periodic or nonperiodic) autocorrelation function and we show an array of constructions, also called multiplications, that give many new infinite families of such sequences. We introduce cyclotomy and show how such sequences together with a search through incidence matrices of cyclotomic or generalised cosets can be used to find a variety of new combinatorial designs. These designs include new weighing matrices, orthogonal designs and D-optimal designs. In particular we give: • T-matrices of order n for n = 13,19,31,37,41,42,43,61,66,86,87. • Weighing matrices W(4n, 4n - 2) and W(4n, 2n - 1) for n = 25,31,37,41,61,71,73,157. • D-optimal designs of order v = 2 = 31,33,37,41,43,61,73,85,91,93,113,145,157,181. We review some of the Stanton-Spratt-Whiteman constructions for difference sets and show how they can be applied to construct supplementary difference sets. We also give many other infinite families of supplementary difference sets and pairwise balanced designs with .A = 1. The thesis concludes with some discussions about low autocorrelation in cryptography and a presentation of a small one-key cryptosystem.