Skip to main content
Erschienen in: Designs, Codes and Cryptography 3/2017

23.01.2016

Fourier transforms and bent functions on faithful actions of finite abelian groups

verfasst von: Yun Fan, Bangteng Xu

Erschienen in: Designs, Codes and Cryptography | Ausgabe 3/2017

Einloggen, um Zugang zu erhalten

Aktivieren Sie unsere intelligente Suche um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

Let G be a finite abelian group acting faithfully on a finite set X. The G-bentness and G-perfect nonlinearity of functions on X are studied by Poinsot and co-authors (Discret Appl Math 157:1848–1857, 2009; GESTS Int Trans Comput Sci Eng 12:1–14, 2005) via Fourier transforms of functions on G. In this paper we introduce the so-called \(G\)-dual set \(\widehat{X}\) of X, which plays the role similar to the dual group \(\widehat{G}\) of G, and develop a Fourier analysis on X, a generalization of the Fourier analysis on the group G. Then we characterize the bentness and perfect nonlinearity of functions on X by their own Fourier transforms on \(\widehat{X}\). Furthermore, we prove that the bentness of a function on X can be determined by its distance from the set of G-linear functions. As direct consequences, many known results in Logachev et al. (Discret Math Appl 7:547–564, 1997), Carlet and Ding (J Complex 20:205–244, 2004), Poinsot (2009), Poinsot et al. (2005) and some new results about bent functions on G are obtained. In order to explain the theory developed in this paper clearly, examples are also presented.
Literatur
1.
Zurück zum Zitat Alperin J.L., Bell R.B.: Groups and Representations, GTM 162. Springer, New York (1997). Alperin J.L., Bell R.B.: Groups and Representations, GTM 162. Springer, New York (1997).
2.
Zurück zum Zitat Carlet C., Ding C.: Highly nonlinear mappings. J. Complex. 20, 205–244 (2004). Carlet C., Ding C.: Highly nonlinear mappings. J. Complex. 20, 205–244 (2004).
3.
Zurück zum Zitat Dillon J.F.: Elementary Hadamard difference sets, Ph.D. Thesis, University of Maryland (1974). Dillon J.F.: Elementary Hadamard difference sets, Ph.D. Thesis, University of Maryland (1974).
4.
Zurück zum Zitat Logachev O.A., Salnikov A.A., Yashchenko V.V.: Bent functions over a finite abelian group. Discret. Math. Appl. 7, 547–564 (1997). Logachev O.A., Salnikov A.A., Yashchenko V.V.: Bent functions over a finite abelian group. Discret. Math. Appl. 7, 547–564 (1997).
5.
Zurück zum Zitat Poinsot L.: Bent functions on a finite nonabelian group. J. Discret. Math. Sci. Cryptogr. 9, 349–364 (2006). Poinsot L.: Bent functions on a finite nonabelian group. J. Discret. Math. Sci. Cryptogr. 9, 349–364 (2006).
6.
Zurück zum Zitat Poinsot L.: A new characterization of group action-based perfect nonlinearity. Discret. Appl. Math. 157, 1848–1857 (2009). Poinsot L.: A new characterization of group action-based perfect nonlinearity. Discret. Appl. Math. 157, 1848–1857 (2009).
7.
Zurück zum Zitat Poinsot L., Harari S.: Group actions based perfect nonlinearity. GESTS Int. Trans. Comput. Sci. Eng. 12, 1–14 (2005). Poinsot L., Harari S.: Group actions based perfect nonlinearity. GESTS Int. Trans. Comput. Sci. Eng. 12, 1–14 (2005).
8.
Zurück zum Zitat Poinsot L., Pott A.: Non-boolean almost perfect nonlinear functions on non-abelian groups. Int. J. Found. Comput. Sci. 22, 1351–1367 (2011). Poinsot L., Pott A.: Non-boolean almost perfect nonlinear functions on non-abelian groups. Int. J. Found. Comput. Sci. 22, 1351–1367 (2011).
9.
Zurück zum Zitat Pott A.: Nonlinear functions in abelian groups and relative difference sets. In: Optimal Discrete Structures and Algorithms, ODSA 2000. Discret. Appl. Math. 138, 177–193 (2004). Pott A.: Nonlinear functions in abelian groups and relative difference sets. In: Optimal Discrete Structures and Algorithms, ODSA 2000. Discret. Appl. Math. 138, 177–193 (2004).
10.
Zurück zum Zitat Rothaus O.S.: On bent functions. J. Comb. Theory A 20, 300–305 (1976). Rothaus O.S.: On bent functions. J. Comb. Theory A 20, 300–305 (1976).
11.
Zurück zum Zitat Serre J.-P.: Representations of Finite Groups. Springer, New York (1984). Serre J.-P.: Representations of Finite Groups. Springer, New York (1984).
12.
Zurück zum Zitat Solodovnikov V.I.: Bent functions from a finite abelian group to a finite abelian group. Diskret. Mat. 14, 99–113 (2002). Solodovnikov V.I.: Bent functions from a finite abelian group to a finite abelian group. Diskret. Mat. 14, 99–113 (2002).
13.
Zurück zum Zitat Xu B.: Multidimensional Fourier transforms and nonlinear functions on finite groups. Linear Algebra Appl. 450, 89–105 (2014). Xu B.: Multidimensional Fourier transforms and nonlinear functions on finite groups. Linear Algebra Appl. 450, 89–105 (2014).
14.
Zurück zum Zitat Xu B.: Dual bent functions on finite groups and \(C\)-algebras. J. Pure Appl. Algebra 220, 1055–1073 (2016). Xu B.: Dual bent functions on finite groups and \(C\)-algebras. J. Pure Appl. Algebra 220, 1055–1073 (2016).
15.
Zurück zum Zitat Xu B.: Bentness and nonlinearity of functions on finite groups. Des. Codes Cryptogr. 76, 409–430 (2015). Xu B.: Bentness and nonlinearity of functions on finite groups. Des. Codes Cryptogr. 76, 409–430 (2015).
Metadaten
Titel
Fourier transforms and bent functions on faithful actions of finite abelian groups
verfasst von
Yun Fan
Bangteng Xu
Publikationsdatum
23.01.2016
Verlag
Springer US
Erschienen in
Designs, Codes and Cryptography / Ausgabe 3/2017
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI
https://doi.org/10.1007/s10623-016-0177-8

Weitere Artikel der Ausgabe 3/2017

Designs, Codes and Cryptography 3/2017 Zur Ausgabe