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Non abelian bent functions. (English) Zbl 1282.11165

Summary: Perfect nonlinear functions from a finite group \(G\) to another one \(H\) are those functions \(f: G\to H\) such that for all nonzero \(\alpha \in G\), the derivative \(d_{a}f: x \mapsto f(ax) f(x)^{-1}\) is balanced. In the case where both \(G\) and \(H\) are abelian groups, \(f: G \to H\) is perfect nonlinear if, and only if, \(f\) is bent, i.e., for all nonprincipal characters \(\chi \) of \(H\), the (discrete) Fourier transform of \(\chi \deg f\) has a constant magnitude equal to \(|G|\). In this paper, using the theory of linear representations, we exhibit similar bentness-like characterizations in the cases where \(G\) and/or \(H\) are (finite) non-abelian groups. Thus we extend the concept of bent functions to the framework of non-abelian groups.

MSC:

11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
06E30 Boolean functions
20D99 Abstract finite groups
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
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[1] Biham, E., Shamir, A.: Differential cryptanalysis of DES-like cryptosystems. J. Cryptol. 4(1), 3–72 (1991) · Zbl 0729.68017 · doi:10.1007/BF00630563
[2] Carlet, C., Ding, C.: Highly nonlinear mappings. J. Complex. 20(2), 205–244 (2004) · Zbl 1053.94011 · doi:10.1016/j.jco.2003.08.008
[3] Dillon, J.F.: Elementary Hadamard difference sets. Ph.D. Thesis, University of Maryland (1974) · Zbl 0346.05003
[4] Goodman, R., Wallach, N.R.: Symmetry, Representations, and Invariants. Graduate Texts in Mathematics, vol. 255. Springer (2009) · Zbl 1173.22001
[5] Hewitt, E., Ross, K.: Abstract Harmonic Analysis, vol. I, 2nd edn. Grundlehren der Mathematischen Wissenschaften. Springer (1979) · Zbl 0416.43001
[6] James, G., Liebeck, M.: Representations and Characters of Groups, 2nd edn. Cambridge University Press, Cambridge (2001) · Zbl 0981.20004
[7] Kosmann-Schwarzbach, Y.: Groupes et Symétries. Les éditions de l’École Polytechnique (French) (2005) · Zbl 1132.20001
[8] Matsui, M.: Linear cryptanalysis method for DES cipher. In: Advances in Cryptology–Eurocrypt ’93, Ser. Lecture Notes in Computer Science, vol. 765, pp. 386–397 (1994) · Zbl 0951.94519
[9] Nyberg, K.: Perfect nonlinear S-boxes. In: Advances in Cryptology–Eurocrypt’91, Ser. Lecture Notes in Computer Science, vol. 547, pp. 378–386 (1992)
[10] Peyré, G.: L’algèbre Discrète de la Transformée de Fourier. Collection Mathématiques à l’Université, Ellipses (French) (2004)
[11] Poinsot, L.: Non linearité parfaite généralisée au sens des actions de groupe, contribution aux fondements de la solidité cryptographique. Ph.D. thesis in Mathematics, University of South Toulon-Var, France (French) (2005). Available on-line at http://tel.archives-ouvertes.fr/docs/00/04/84/86/46/PDF/tel-00010216.pdf
[12] Poinsot, L.: Multidimensional bent functions. In: GESTS International Transactions in Computer Science and Engineering, vol. 18(1), pp. 185–195 (2005)
[13] Poinsot, L.: A new characterization of group action-based perfect nonlinearity. Discrete Appl. Math. 157(8), 1848–1857 (2009) · Zbl 1166.94007 · doi:10.1016/j.dam.2009.02.001
[14] Pott, A.: Nonlinear functions in Abelian groups and relative difference sets. Discrete Appl. Math. 138(1–2), 177–193 (2004) · Zbl 1035.05023 · doi:10.1016/S0166-218X(03)00293-2
[15] Rothaus, O.S.: On bent functions. J. Comb. Theory A 20, 300–365 (1976) · Zbl 0336.12012 · doi:10.1016/0097-3165(76)90024-8
[16] Serre, J.-P.: Linear Representations of Finite Groups. Graduate Texts in Mathematics, vol. 42, Springer (1977)
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