Cryptography and Communications

, Volume 11, Issue 6, pp 1233–1245

Generalized bent functions into $$\mathbb {Z}_{p^{k}}$$ from the partial spread and the Maiorana-McFarland class

• Wilfried Meidl
• Alexander Pott
Article
Part of the following topical collections:
1. Special Issue on Boolean Functions and Their Applications

Abstract

Functions f from $${\mathbb {F}_{p}^{n}}$$, n = 2m, to $$\mathbb {Z}_{{p}^{k}}$$ for which the character sum $$\mathcal {H}^{k}_{f}(p^{t},u)=\sum\limits _{x\in {\mathbb {F}_{p}^{n}}}\zeta _{p^{k}}^{p^{t}f(x)}\zeta _{p}^{u\cdot x}$$ (where $$\zeta _{q} = e^{2\pi i/q}$$ is a q-th root of unity), has absolute value $$p^{m}$$ for all $$u\in {\mathbb {F}_{p}^{n}}$$ and $$0\le t\le k-1$$, induce relative difference sets in $${\mathbb {F}_{p}^{n}}\times \mathbb {Z}_{{p}^{k}}$$ hence are called bent. Functions only necessarily satisfying $$|\mathcal {H}^{k}_{f}(1,u)| = p^{m}$$ are called generalized bent. We show that with spreads we not only can construct a variety of bent and generalized bent functions, but also can design functions from $${\mathbb {F}_{p}^{n}}$$ to $$\mathbb {Z}_{{p}^{m}}$$ satisfying $$|\mathcal {H}_{f}^{m}(p^{t},u)| = p^{m}$$ if and only if $$t\in T$$ for any $$T\subset \{0,1\ldots ,m-1\}$$. A generalized bent function can also be seen as a Boolean (p-ary) bent function together with a partition of $${\mathbb {F}_{p}^{n}}$$ with certain properties. We show that the functions from the completed Maiorana-McFarland class are bent functions, which allow the largest possible partitions.

Keywords

Bent function Generalized bent function Partial spread Maiorana-McFarland Walsh transform Relative difference set

Mathematics Subject Classification (2010)

06E30 05B10 94C10

Notes

Acknowledgments

W.M. is supported by the FWF Project P 30966.

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