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 Diskretn. Anal. Issled. Oper., 2012, Volume 19, Number 1, Pages 41–58 (Mi da676)

Enumeration of bent functions on the minimal distance from the quadratic bent function

N. A. Kolomeec

S. L. Sobolev Institute of Mathematics, SB RAS, Novosibirsk, Russia

Abstract: Constructing bent functions on the minimal distance from the quadratic bent function is studied. All such bent functions in $2k$ variables are obtained and it is shown that the number of them is equal to $2^k(2^1+1)…(2^k+1)$. A lower bound of the number of bent functions on the minimal distance from a Maiorana–McFarland bent function is given. Tab. 1, bibliogr. 9.

Keywords: bent function, the minimal distance, quadratic bent function.

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English version:
Journal of Applied and Industrial Mathematics, 2012, 6:3, 306–317

Bibliographic databases:

UDC: 519.7
Revised: 24.09.2011

Citation: N. A. Kolomeec, “Enumeration of bent functions on the minimal distance from the quadratic bent function”, Diskretn. Anal. Issled. Oper., 19:1 (2012), 41–58; J. Appl. Industr. Math., 6:3 (2012), 306–317

Citation in format AMSBIB
\Bibitem{Kol12} \by N.~A.~Kolomeec \paper Enumeration of bent functions on the minimal distance from the quadratic bent function \jour Diskretn. Anal. Issled. Oper. \yr 2012 \vol 19 \issue 1 \pages 41--58 \mathnet{http://mi.mathnet.ru/da676} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2961451} \transl \jour J. Appl. Industr. Math. \yr 2012 \vol 6 \issue 3 \pages 306--317 \crossref{https://doi.org/10.1134/S1990478912030052} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. G. I. Shushuev, “Vektornye bulevy funktsii na rasstoyanii odin ot APN-funktsii”, PDM. Prilozhenie, 2014, no. 7, 36–37
2. N. A. Kolomeets, “Verkhnyaya otsenka chisla bent-funktsii na rasstoyanii $2^k$ ot proizvolnoi bent-funktsii ot $2k$ peremennykh”, PDM, 2014, no. 3(25), 28–39
3. V. N. Potapov, “Svoistva $p$-ichnykh bent-funktsii, nakhodyaschikhsya na minimalnom rasstoyanii drug ot druga”, PDM. Prilozhenie, 2015, no. 8, 39–43
4. N. Kolomeec, “The graph of minimal distances of bent functions and its properties”, Des. Codes Cryptogr., 85:3 (2017), 395–410
5. Y. Zhao, N. Cao, Zh. Qi, G. Li, P. Liu, “A new algorithm for enumerating bent functions based on truth tables and run length”, IEEE Access, 6 (2018), 23800–23805
6. Zhao Y., Zhang F., Qi Ch., “A Novel Algorithm Enumerating Bent Functions Based on Value Distribution and Run Length”, Lecture Notes in Real-Time Intelligent Systems (Rtis 2016), Advances in Intelligent Systems and Computing, 613, eds. MizeraPietraszko J., Pichappan P., Springer International Publishing Ag, 2018, 250–259
7. Meidl W., Pott A., “Functions F From Fn P, N=2M, to Z Pk For Which the Character Sum Hk F (Pt, U) = Similar to X.Fn P. Ptf (X) Pk. U. X P (Where.Q=E2Pi/ Q Is a Q-Th Root of Unity), Has Absolute Value Pm For All U. Fn P and 0=T = K-1, Induce Relative Difference Sets in Fn P X Z Pk Hence Are Called Bent. Functions Only Necessarily Satisfying |Hk F (1, U)| = Pm Are Called Generalized Bent. We Show That With Spreads We Not Only Can Construct a Variety of Bent and Generalized Bent Functions, But Also Can Design Functions From Fn P to Zpm Satisfying |Hm F (Pt, U)| = Pm If and Only If T. T For Any T. (0, 1..., M-1). a Generalized Bent Function Can Also Be Seen as a Boolean (P-Ary) Bent Function Together With a Partition of Fn P With Certain Properties. We Show That the Functions From the Completed Maiorana-Mcfarland Class Are Bent Functions, Which Allow the Largest Possible Partitions.”, Cryptogr. Commun., 11:6, SI (2019), 1233–1245
8. Luo G., Cao X., Mesnager S., “Several New Classes of Self-Dual Bent Functions Derived From Involutions”, Cryptogr. Commun., 11:6, SI (2019), 1261–1273
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