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Designs, Codes and Cryptography
Published in: Designs, Codes and Cryptography 2/2016

01-08-2016

Switchings of semifield multiplications

Authors: Xiang-dong Hou, Ferruh Özbudak, Yue Zhou

Published in: Designs, Codes and Cryptography | Issue 2/2016

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Abstract

Let \(B(X,Y)\) be a polynomial over \(\mathbb {F}_{q^n}\) which defines an \(\mathbb {F}_q\)-bilinear form on the vector space \(\mathbb {F}_{q^n}\), and let \(\xi \) be a nonzero element in \(\mathbb {F}_{q^n}\). In this paper, we consider for which \(B(X,Y)\), the binary operation \(xy+B(x,y)\xi \) defines a (pre)semifield multiplication on \({\mathbb {F}}_{q^n}\). We prove that this question is equivalent to finding \(q\)-linearized polynomials \(L(X)\in \mathbb {F}_{q^n}[X]\) such that \( {\mathrm {Tr}}_{q^n/q}(L(x)/x)\ne 0\) for all \(x\in \mathbb {F}_{q^n}^*\). For \(n\le 4\), we present several families of \(L(X)\) and we investigate the derived (pre)semifields. When \(q\) equals a prime \(p\), we show that if \(n>\frac{1}{2}(p-1)(p^2-p+4)\), \(L(X)\) must be \(a_0 X\) for some \(a_0\in \mathbb {F}_{p^n}\) satisfying \( {\mathrm {Tr}}_{q^n/q}(a_0)\ne 0\). Finally, we include a natural connection with certain cyclic codes over finite fields, and we apply the Hasse–Weil–Serre bound for algebraic curves to prove several necessary conditions for such kind of \(L(X)\).
next article On Jacobi sums, difference sets and partial difference sets in Galois domains
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Metadata
Title
Switchings of semifield multiplications
Authors
Xiang-dong Hou
Ferruh Özbudak
Yue Zhou
Publication date
01-08-2016
Publisher
Springer US
Published in
Designs, Codes and Cryptography / Issue 2/2016
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI
https://doi.org/10.1007/s10623-015-0081-7

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