γ-BUTSON-HADAMARD MATRICES AND THEIR CRYPTOGRAPHIC APPLICATIONS

Abstract

A Hadamard matrix is a square matrix with entries ±1 whose rows are orthogonal to each other. Hadamard matrices appear in various fields including cryptography, coding theory, combinatorics etc. This thesis takes an interest in γ-Butson-Hadamard matrix that is a generalization of Hadamard matrices for γ ∈ R ∩ Z[ζm]. These matrices are examined for non-existence cases in this thesis. In particular, the unsolvability of certain equations is studied in the case of cyclotomic number fields whose ring of integers is not a principal ideal domain. Winterhof et al. considered the equations for γ ∈ Z. We first extend this result to γ ∈ R ∩ Z[ζm] by using some new methods from algebraic number theory. Secondly, we obtain another method for checking the non-existence cases of these equations, which uses the tool of norm from algebraic number theory. Then, the direct applications of these results to γ-Butson-Hadamard matrices, γ-Conference matrices, nearly perfect sequences are obtained. Finally, the connection between nonlinear Boolean cryptographic functions and γ-Butson-Hadamard matrices having small |γ| is established. In addition, a computer search is done for checking the cases which are excluded by our results and for obtaining new examples of existence parameters.

Keywords: Butson-Hadamard matrices, algebraic number fields, nearly perfect sequences, conference matrices, cryptographic functions