### An Open Problem on the Bentness of Mesnager’s Functions

Chunming Tang, Peng Han, Qi Wang, Jun Zhang, and Yanfeng Qi

##### Abstract

Let $n=2m$. In the present paper, we study the binomial Boolean functions of the form $$f_{a,b}(x) = \mathrm{Tr}_1^{n}(a x^{2^m-1 }) +\mathrm{Tr}_1^{2}(bx^{\frac{2^n-1}{3} }),$$ where $m$ is an even positive integer, $a\in \mathbb{F}_{2^n}^*$ and $b\in \mathbb{F}_4^*$. We show that $f_{a,b}$ is a bent function if the Kloosterman sum $$K_{m}\left(a^{2^m+1}\right)=1+ \sum_{x\in \mathbb{F}_{2^m}^*} (-1)^{\mathrm{Tr}_1^{m}(a^{2^m+1} x+ \frac{1}{x})}$$ equals $4$, thus settling an open problem of Mesnager. The proof employs tools including computing Walsh coefficients of Boolean functions via multiplicative characters, divisibility properties of Gauss sums, and graph theory.

Available format(s)
Category
Foundations
Publication info
Preprint. Minor revision.
Keywords
Boolean functionbent functionWalsh transformGauss Sumdirected graph
Contact author(s)
tangchunmingmath @ 163 com
History
Short URL
https://ia.cr/2021/1304

CC BY

BibTeX

@misc{cryptoeprint:2021/1304,
author = {Chunming Tang and Peng Han and Qi Wang and Jun Zhang and Yanfeng Qi},
title = {An Open Problem on the Bentness of Mesnager’s Functions},
howpublished = {Cryptology ePrint Archive, Paper 2021/1304},
year = {2021},
note = {\url{https://eprint.iacr.org/2021/1304}},
url = {https://eprint.iacr.org/2021/1304}
}

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