Near Butson-Hadamard Matrices and Nonlinear Boolean Functions

Abstract

A Hadamard matrix is a square matrix with entries $\pm 1$ whose rows are orthogonal to each other. Hadamard matrices appear in various fields including cryptography, coding theory, combinatorics etc. This study takes an interest in $\gamma$ near Butson-Hadamard matrix that is a generalization of Hadamard matrices for $\gamma \in {\mathbb R}\cap {\mathbb Z}[\zeta _m]$. These matrices are examined in this study. In particular, the unsolvability of certain equations is studied in the case of cyclotomic number fields. Winterhof et al. considered the equations for $\gamma \in {\mathbb Z}$, and by the authors for $\gamma \in {\mathbb R}\cap {\mathbb Z}[\zeta _m]$. In this study, we obtain another method for checking the nonexistence cases of these equations, which uses the tool of norm from algebraic number theory. Then, the direct applications of these results to $\gamma$ near Butson-Hadamard matrices are obtained. In the second part of this study, the connection between nonlinear Boolean cryptographic functions and $\gamma$ near Butson-Hadamard matrices having small $|\gamma |$ 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.