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Problemy Peredachi Informatsii, 2008, Volume 44, Issue 1, Pages 15–37
(Mi ppi1263)
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This article is cited in 14 scientific papers (total in 14 papers)
Coding Theory
Bent and Hyper-bent Functions over a Field of $2^l$ Elements
A. S. Kuz'mina, V. T. Markovb, A. A. Nechaevb, V. A. Shishkina, A. B. Shishkova a Moscow Institute of Radio-Engineering, Electronics and Automation
b M. V. Lomonosov Moscow State University
Abstract:
We study the parameters of bent and hyper-bent (HB) functions in n variables over a field $P=\mathbb{F}_q$ with $q=2^l$ elements, $l>1$. Any such function is identified with a function $F:Q\to P$, where $P<Q=\mathbb{F}_qn$. The latter has a reduced trace representation $F=\mathrm{tr}^Q_P(\Phi)$, where $\Phi(x)$ is a uniquely defined polynomial of a special type. It is shown that the most accurate generalization of results on parameters of bent functions from the case $l=1$ to the case $l>1$ is obtained if instead of the nonlinearity degree of a function one considers its binary nonlinearity index (in the case $l=1$ these parameters coincide). We construct a class of HB functions that generalize binary HB functions found in [Youssef, A. M. and Gong, G., Lect. Notes Comp. Sci., vol. 2045, Berlin: Springer, 2001, pp. 406–419]; we indicate a set of parameters $q$ and $n$ for which there are no other HB functions. We introduce the notion of the period of a function and establish a relation between periods of (hyper-)bent functions and their frequency characteristics.
Received: 18.06.2007 Revised: 18.12.2007
Citation:
A. S. Kuz'min, V. T. Markov, A. A. Nechaev, V. A. Shishkin, A. B. Shishkov, “Bent and Hyper-bent Functions over a Field of $2^l$ Elements”, Probl. Peredachi Inf., 44:1 (2008), 15–37; Problems Inform. Transmission, 44:1 (2008), 12–33
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https://www.mathnet.ru/eng/ppi1263 https://www.mathnet.ru/eng/ppi/v44/i1/p15
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Abstract page: | 780 | Full-text PDF : | 255 | References: | 64 | First page: | 11 |
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