Paper 2011/698

A generalization of the class of hyper-bent Boolean functions in binomial forms

Chunming Tang, Yu Lou, Yanfeng Qi, Baocheng Wang, and Yixian Yang

Abstract

Bent functions, which are maximally nonlinear Boolean functions with even numbers of variables and whose Hamming distance to the set of all affine functions equals $2^{n-1}\pm 2^{\frac{n}{2}-1}$, were introduced by Rothaus in 1976 when he considered problems in combinatorics. Bent functions have been extensively studied due to their applications in cryptography, such as S-box, block cipher and stream cipher. Further, they have been applied to coding theory, spread spectrum and combinatorial design. Hyper-bent functions, as a special class of bent functions, were introduced by Youssef and Gong in 2001, which have stronger properties and rarer elements. Many research focus on the construction of bent and hyper-bent functions. In this paper, we consider functions defined over $\mathbb{F}_{2^n}$ by $f^{(r)}_{a,b}:=\mathrm{Tr}_{1}^{n}(ax^{r(2^m-1)}) +\mathrm{Tr}_{1}^{4}(bx^{\frac{2^n-1}{5}})$, where $n=2m$, $m\equiv 2\pmod 4$, $a\in \mathbb{F}_{2^m}$ and $b\in\mathbb{F}_{16}$. When $r\equiv 0\pmod 5$, we characterize the hyper-bentness of $f^{(r)}_{a,b}$. When $r\not \equiv 0\pmod 5$, $a\in mathbb{F}_{2^m}$ and $(b+1)(b^4+b+1)=0$, with the help of Kloosterman sums and the factorization of $x^5+x+a^{-1}$, we present a characterization of hyper-bentness of $f^{(r)}_{a,b}$. Further, we give all the hyper-bent functions of $f^{(r)}_{a,b}$ in the case $a\in\mathbb{F}_{2^{\frac{m}{2}}}$.

Metadata
Available format(s)
PDF
Publication info
Published elsewhere. Unknown where it was published
Keywords
Boolean functionsbent functionshyper-bent functionsWalsh-Hadamard transformationKloosterman sums
Contact author(s)
tangchunmingmath @ 163 com
History
2012-11-12: last of 3 revisions
2011-12-23: received
See all versions
Short URL
https://ia.cr/2011/698
License
Creative Commons Attribution
CC BY

BibTeX

@misc{cryptoeprint:2011/698,
      author = {Chunming Tang and Yu Lou and Yanfeng Qi and Baocheng Wang and Yixian Yang},
      title = {A generalization of the class of  hyper-bent Boolean functions in binomial forms},
      howpublished = {Cryptology ePrint Archive, Paper 2011/698},
      year = {2011},
      note = {\url{https://eprint.iacr.org/2011/698}},
      url = {https://eprint.iacr.org/2011/698}
}
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