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Erschienen in:
21.05.2019
Generalized bent functions into \(\mathbb {Z}_{p^{k}}\) from the partial spread and the Maiorana-McFarland class
Erschienen in: Cryptography and Communications | Ausgabe 6/2019
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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.