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Bent and hyper-bent functions over a field of $$2^{\ell}$$ elements. (English. Russian original) Zbl 1235.94049
Probl. Inf. Transm. 44, No. 1, 12-33 (2008); translation from Probl. Peredachi Inf. 44, No. 1, 15-37 (2008).
Summary: We study the parameters of bent and hyper-bent (HB) functions in $$n$$ variables over a field $$P = \mathbb{F}_q$$ with $$q = 2^{\ell}$$ elements, $$\ell > 1$$. Any such function is identified with a function $$F: Q \to P$$, where $$P < Q = \mathbb{F}_{q^n}$$. The latter has a reduced trace representation $$F = \text{tr}_P^Q (\Phi)$$, where $$\Phi(x)$$ is a uniquely defined polynomial of a special type. It is shown that the most accurate generalization of results on parameters of bent functions from the case $$\ell = 1$$ to the case $$\ell > 1$$ is obtained if instead of the nonlinearity degree of a function one considers its binary nonlinearity index (in the case $$\ell = 1$$ these parameters coincide). We construct a class of HB functions that generalize binary HB functions found in [A. M. Youssef and G. Gong, Advances in cryptology – EUROCRYPT 2001, Lect. Notes Comput. Sci. 2045, 406–419 (2001; Zbl 1013.94544)]; we indicate a set of parameters $$q$$ and $$n$$ for which there are no other HB functions. We introduce the notion of the period of a function and establish a relation between periods of (hyper-)bent functions and their frequency characteristics.

##### MSC:
 94A60 Cryptography 11T71 Algebraic coding theory; cryptography (number-theoretic aspects) 11T06 Polynomials over finite fields 06E30 Boolean functions 94C10 Switching theory, application of Boolean algebra; Boolean functions (MSC2010)
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##### References:
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