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Designs, Codes and Cryptography

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27.05.2016

On some permutation binomials and trinomials over \(\mathbb {F}_{2^n}\)

verfasst von: Srimanta Bhattacharya, Sumanta Sarkar

Erschienen in: Designs, Codes and Cryptography | Ausgabe 1-2/2017

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Abstract

In this work, we completely characterize (1) permutation binomials of the form \(x^{{{2^n -1}\over {2^t-1}}+1}+ ax \in \mathbb {F}_{2^n}[x], n = 2^st, a \in \mathbb {F}_{2^{2t}}^{*}\), and (2) permutation trinomials of the form \(x^{2^s+1}+x^{2^{s-1}+1}+\alpha x \in \mathbb {F}_{2^t}[x]\), where s, t are positive integers. The first result, which was our primary motivation, is a consequence of the second result. The second result may be of independent interest.

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These are polynomials of the form \(x^rf(x^{q-1\over d})\), and represent mappings of the factor group \(\mathbb {F}_{q}^{*} / C_d\) to itself, where \(C_d\) is the subgroup of \(\mathbb {F}_{q}^{*}\) of index d (see [9] for further details). We review relevant characterization results of this class of binomials in Sect. 1.2.

A polynomial \(f(x)\in \mathbb {F}_q[x]\) is called complete mapping if both f(x) and \(f(x)+x\) are PPs of \(\mathbb {F}_q\), and orthomorphism if both f(x) and \(f(x)-x\) are PPs; for even characteristic both are same. Complete mappings/orthomorphisms are useful for construction of mutually orthogonal latin squares (see [9, 26]).

The case of \(d=2\) was settled in [18]. However, it is relevant for fields of odd characteristic.

In [22, 23], the author characterized these PBs for any d in terms of Lucas sequences. However, as we have stated before, we are interested in more explicit characterization.

PBs of the form \(x^{2{q^2-1\over q-1}+1}+ax\) over \(\mathbb {F}_{q^2}\) were also characterized in the same work.

This approach was taken in [24, 26] (see also [2] and references therein).

Conditions (b) and (c) can be written together (see [25]) as the condition: \(x^rf(x)^{q-1\over d}\) permutes the set \(\{a \in \mathbb {F}_q: a^d=1\}.\)

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Titel: On some permutation binomials and trinomials over
verfasst von: Srimanta Bhattacharya
Sumanta Sarkar
Publikationsdatum: 27.05.2016
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 1-2/2017
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-016-0229-0

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On some permutation binomials and trinomials over \(\mathbb {F}_{2^n}\)

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