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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alperin J.L., Bell R.B.: Groups and Representations, GTM 162. Springer, New York (1997).
Carlet C., Ding C.: Highly nonlinear mappings. J. Complex. 20, 205–244 (2004).
Dillon J.F.: Elementary Hadamard difference sets, Ph.D. Thesis, University of Maryland (1974).
Logachev O.A., Salnikov A.A., Yashchenko V.V.: Bent functions over a finite abelian group. Discret. Math. Appl. 7, 547–564 (1997).
Poinsot L.: Bent functions on a finite nonabelian group. J. Discret. Math. Sci. Cryptogr. 9, 349–364 (2006).
Poinsot L.: A new characterization of group action-based perfect nonlinearity. Discret. Appl. Math. 157, 1848–1857 (2009).
Poinsot L., Harari S.: Group actions based perfect nonlinearity. GESTS Int. Trans. Comput. Sci. Eng. 12, 1–14 (2005).
Poinsot L., Pott A.: Non-boolean almost perfect nonlinear functions on non-abelian groups. Int. J. Found. Comput. Sci. 22, 1351–1367 (2011).
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).
Rothaus O.S.: On bent functions. J. Comb. Theory A 20, 300–305 (1976).
Serre J.-P.: Representations of Finite Groups. Springer, New York (1984).
Solodovnikov V.I.: Bent functions from a finite abelian group to a finite abelian group. Diskret. Mat. 14, 99–113 (2002).
Xu B.: Multidimensional Fourier transforms and nonlinear functions on finite groups. Linear Algebra Appl. 450, 89–105 (2014).
Xu B.: Dual bent functions on finite groups and \(C\)-algebras. J. Pure Appl. Algebra 220, 1055–1073 (2016).
Xu B.: Bentness and nonlinearity of functions on finite groups. Des. Codes Cryptogr. 76, 409–430 (2015).
Acknowledgments
This work was done while the first author was visiting the second author at Eastern Kentucky University in Spring 2014; he is grateful for the hospitality. The work of the first author is supported by NSFC with Grant Number 11271005. The authors would like to thank the referees for their useful comments, especially for the suggestions of Theorem 6.2 and Example 6.6.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Pott.
Rights and permissions
About this article
Cite this article
Fan, Y., Xu, B. Fourier transforms and bent functions on faithful actions of finite abelian groups. Des. Codes Cryptogr. 82, 543–558 (2017). https://doi.org/10.1007/s10623-016-0177-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-016-0177-8
Keywords
- Group actions
- G-linear functions
- \(G\)-dual sets
- Fourier transforms on G-sets
- Bent functions
- Perfect nonlinear functions