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Designs, Codes and Cryptography

, Volume 80, Issue 2, pp 217–239 | Cite as

Switchings of semifield multiplications

  • Xiang-dong Hou
  • Ferruh Özbudak
  • Yue Zhou
Article

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 \(L(X)\).

Keywords

Cyclic code Finite field Linearized polynomial  Semifield  The Hasse–Weil–Serre bound 

Mathematics Subject Classification

11T55 12E20 12K10 14H05 94B15 

Notes

Acknowledgments

The authors are very grateful to the anonymous referees for their valuable comments and suggestions. Xiang-dong Hou is research partially supported by NSA Grant H98230-12-1-0245. Ferruh Özbudak is research partially supported by TUBİTAK under Grant No. TBAG-112T011. Yue Zhou is partially supported by the National Basic Research Program of China (No. 2013CB338002) and the National Natural Science Foundation of China (No. 11401579, 61272484).

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of South FloridaTampaUSA
  2. 2.Department of Mathematics and the Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey
  3. 3.College of ScienceNational University of Defense TechnologyChangshaChina

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