A construction of bent functions with optimal algebraic degree and large symmetric group

Wenying Zhang, Zhaohui Xing, Keqin Feng

Abstract: As maximal, nonlinear Boolean functions, bent functions have many theoretical and practical applications in combinatorics, coding theory, and cryptography. In this paper, we present a construction of bent function $ f_{a,S} $ with $ n = 2m $ variables for any nonzero vector $ a\in \mathbb{F}_{2}^{m} $ and subset $ S $ of $ \mathbb{F}_{2}^{m} $ satisfying $ a+S = S $. We give a simple expression of the dual bent function of $ f_{a,S} $ and prove that $ f_{a,S} $ has optimal algebraic degree $ m $ if and only if $ |S|\equiv 2 (\bmod 4)  $. This construction provides a series of bent functions with optimal algebraic degree and large symmetric group if $ a $ and $ S $ are chosen properly. We also give some examples of those bent functions $ f_{a,S} $ and their dual bent functions. Keywords: Boolean functions, bent functions, dual functions, algebraic degree, symmetric group. Mathematics Subject Classification: 03G05, 06E25. Citation: \begin{equation} \\ \end{equation}