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Probl. Peredachi Inf., 2008, Volume 44, Issue 1, Pages 15–37 (Mi ppi1263)  

This article is cited in 5 scientific papers (total in 5 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.

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English version:
Problems of Information Transmission, 2008, 44:1, 12–33

Bibliographic databases:

UDC: 621.391.15:519.1
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

Citation in format AMSBIB
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\paper Bent and Hyper-bent Functions over a~Field of $2^l$ Elements
\jour Probl. Peredachi Inf.
\yr 2008
\vol 44
\issue 1
\pages 15--37
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\jour Problems Inform. Transmission
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\vol 44
\issue 1
\pages 12--33
\crossref{https://doi.org/10.1134/S003294600801002X}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. N. N. Tokareva, “On Quadratic Approximations in Block Ciphers”, Problems Inform. Transmission, 44:3 (2008), 266–286  mathnet  crossref  mathscinet  zmath  isi
    2. N. N. Tokareva, “Bent-funktsii: rezultaty i prilozheniya. Obzor rabot”, PDM, 2009, no. 1(3), 15–37  mathnet
    3. N. N. Tokareva, “Generalizations of bent functions”, J. Appl. Industr. Math., 5:1 (2011), 110–129  mathnet  crossref  mathscinet  zmath
    4. V. A. Shishkin, “Nekotorye svoistva $q$-ichnykh bent-funktsii”, PDM. Prilozhenie, 2014, no. 7, 33–34  mathnet
    5. A. D. Bugrov, “Kusochno-affinnye podstanovki konechnykh polei”, PDM, 2015, no. 4(30), 5–23  mathnet  crossref
  • Проблемы передачи информации Problems of Information Transmission
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