Switchings of semifield multiplications


Creative Commons License

Hou X., ÖZBUDAK F., ZHOU Y.

DESIGNS CODES AND CRYPTOGRAPHY, vol.80, no.2, pp.217-239, 2016 (Peer-Reviewed Journal) identifier identifier

  • Publication Type: Article / Article
  • Volume: 80 Issue: 2
  • Publication Date: 2016
  • Doi Number: 10.1007/s10623-015-0081-7
  • Journal Name: DESIGNS CODES AND CRYPTOGRAPHY
  • Journal Indexes: Science Citation Index Expanded, Scopus
  • Page Numbers: pp.217-239

Abstract

Let B(X, Y) be a polynomial over F-qn which defines an F-q-bilinear form on the vector space F-qn, and let xi be a nonzero element in F-qn. In this paper, we consider for which B(X, Y), the binary operation xy + B(x, y) xi defines a (pre)semifield multiplication on F-qn. We prove that this question is equivalent to finding q-linearized polynomials L(X) is an element of F-qn [X] such that Tr-qn/q (L(x)/x) not equal 0 for all x is an element of F-qn*. For n <= 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 > 1/2(p - 1)(p(2) - p + 4), L(X) must be a(0)X for some a(0) is an element of F-pn satisfying Tr-qn/q (a(0)) not equal 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).