Paper 2019/307
Solving $x^{2^k+1}+x+a=0$ in $\mathbb{F}_{2^n}$ with $\gcd(n,k)=1$
Kwang Ho Kim and Sihem Mesnager
Abstract
Let $N_a$ be the number of solutions to the equation $x^{2^k+1}+x+a=0$ in $\mathbb F_{n}$ where $\gcd(k,n)=1$. In 2004, by Bluher it was known that possible values of $N_a$ are only 0, 1 and 3. In 2008, Helleseth and Kholosha have got criteria for $N_a=1$ and an explicit expression of the unique solution when $\gcd(k,n)=1$. In 2014, Bracken, Tan and Tan presented a criterion for $N_a=0$ when $n$ is even and $\gcd(k,n)=1$. This paper completely solves this equation $x^{2^k+1}+x+a=0$ with only condition $\gcd(n,k)=1$. We explicitly calculate all possible zeros in $\mathbb F_{n}$ of $P_a(x)$. New criterion for which $a$, $N_a$ is equal to $0$, $1$ or $3$ is a byproduct of our result.
Metadata
 Available format(s)
 Category
 Foundations
 Publication info
 Preprint. Minor revision.
 Keywords
 MullerCohenMatthewspolynomials Dickson polynomialZeros of polynomialIrreducible polynomials
 Contact author(s)
 smesnager @ univparis8 fr
 History
 20190320: 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}, note = {\url{https://eprint.iacr.org/2019/307}}, url = {https://eprint.iacr.org/2019/307} }