Solving $x^{2^k+1}+x+a=0$ in $\mathbb{F}_{2^n}$ with $\gcd(n,k)=1$

Kwang Ho Kim; Sihem Mesnager

Paper 2019/307

Solving in with

Kwang Ho Kim and Sihem Mesnager

Abstract

Let be the number of solutions to the equation in where . In 2004, by Bluher it was known that possible values of are only 0, 1 and 3. In 2008, Helleseth and Kholosha have got criteria for and an explicit expression of the unique solution when . In 2014, Bracken, Tan and Tan presented a criterion for when is even and . This paper completely solves this equation with only condition . We explicitly calculate all possible zeros in of . New criterion for which , is equal to , or is a by-product of our result.

Metadata

Available format(s): PDF
Category: Foundations
Publication info: Preprint. MINOR revision.
Keywords: Muller-Cohen-Matthews polynomials Dickson polynomial Zeros of polynomial Irreducible polynomials
Contact author(s): smesnager @ univ-paris8 fr
History: 2019-03-20: received
Short URL: https://ia.cr/2019/307
License: CC BY

BibTeX

@misc{cryptoeprint:2019/307,
      author = {Kwang Ho Kim and Sihem Mesnager},
      title = {Solving $x^{2^k+1}+x+a=0$ in $\mathbb{F}_{2^n}$ with $\gcd(n,k)=1$},
      howpublished = {Cryptology {ePrint} Archive, Paper 2019/307},
      year = {2019},
      url = {https://eprint.iacr.org/2019/307}
}