Paper 2021/1304
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 $$f_{a,b}(x) = \mathrm{Tr}_1^{n}(a x^{2^m1 }) +\mathrm{Tr}_1^{2}(bx^{\frac{2^n1}{3} }), $$ 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.
Metadata
 Available format(s)
 Category
 Foundations
 Publication info
 Preprint. Minor revision.
 Keywords
 Boolean functionbent functionWalsh transformGauss Sumdirected graph
 Contact author(s)
 tangchunmingmath @ 163 com
 History
 20210928: received
 Short URL
 https://ia.cr/2021/1304
 License

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} }