Cryptology ePrint Archive: Report 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 by-product of our result.

Category / Keywords: foundations / Muller-Cohen-Matthews; polynomials Dickson polynomial; Zeros of polynomial; Irreducible polynomials

Date: received 18 Mar 2019

Contact author: smesnager at univ-paris8 fr

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Version: 20190320:103458 (All versions of this report)

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