Vectorial Hyperbent Trace Functions From the \(\mathcal {PS}_{\rm ap}\) Class—Their Exact Number and Specification
| Autor: | Enes Pasalic, Amela Muratovic-Ribic, Samir Ribic |
|---|---|
| Rok vydání: | 2014 |
| Předmět: |
Discrete mathematics
Trace (linear algebra) Root of unity Bent molecular geometry Lagrange polynomial Cyclic group Library and Information Sciences Computer Science Applications Polynomial interpolation Combinatorics symbols.namesake symbols Boolean function Information Systems Mathematics Interpolation |
| Zdroj: | IEEE Transactions on Information Theory. 60:4408-4413 |
| ISSN: | 1557-9654 0018-9448 |
| DOI: | 10.1109/tit.2014.2320269 |
| Popis: | To identify and specify trace bent functions of the form \(Tr(P(x))\) , where \(P(x) \in {\mathbb F} _{2^n}[x]\) , has been an important research topic lately. We characterize a class of vectorial (hyper)bent functions of the form \(F(x)=Tr_k^n(\sum _{i=0}^{2^k}a_ix^{i(2^k-1)})\) , where \(n=2k\) , in terms of finding an explicit expression for the coefficients \(a_i\) so that \(F\) is vectorial hyperbent. These coefficients only depend on the choice of the interpolating polynomial used in the Lagrange interpolation of the elements of \(\mathcal {U}\) and some prespecified outputs, where \(\mathcal {U}\) is the cyclic group of \((2^{n/2}+1)\) th roots of unity in \( {\mathbb F}_{2^n}\) . We show that these interpolation polynomials can be chosen in exactly \((2^k+1)!2^{k-1}\) ways and this is the exact number of vectorial hyperbent functions of the form \(Tr^n_k(\sum _{i=0}^{2^k}a_ix^{i(2^k-1)})\) . Furthermore, a simple optimization method is proposed for selecting the interpolation polynomials that give rise to trace polynomials with a few nonzero coefficients. |
| Databáze: | OpenAIRE |
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