On Generalized Bent and q-ary Perfect Nonlinear Functions

Claude Carlet () and Sylvie Dubuc ()
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Claude Carlet: Université de Caen, GREYC
Sylvie Dubuc: Université de Caen, GREYC

A chapter in Finite Fields and Applications, 2001, pp 81-94 from Springer

Abstract: Abstract The notion of bent function has been generalized by Kumar et al. to the alphabet = Z q = (q Z) and also studied by Nyberg. The classical equivalent definitions of binary bent functions lead, through this generalization, to the notions of generalized bent functions and of q-ary perfect nonlinear functions. We show in this paper that a q-ary function f is perfect nonlinear if and only if, for every u in Z q * , the function uf is generalized bent. We check that, among all known constructions of generalized bent functions, only one (due to Hou) can produce perfect nonlinear functions. This construction works for n even (n > 2) and the question of knowing whether there exist perfect nonlinear functions for n odd arises. We introduce a construction of perfect nonlinear functions on ℤ4 n , for every n > 1.

Date: 2001
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DOI: 10.1007/978-3-642-56755-1_8

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