Advertisement
Published in:
21-05-2019
Generalized bent functions into \(\mathbb {Z}_{p^{k}}\) from the partial spread and the Maiorana-McFarland class
Published in: Cryptography and Communications | Issue 6/2019
Log inActivate our intelligent search to find suitable subject content or patents.
Select sections of text to find matching patents with Artificial Intelligence. powered by
Select sections of text to find additional relevant content using AI-assisted search. powered by
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.