On the Equation $x^{2^l+1}+x+a=0$ over $\mathrm{GF}(2^k)$ (Extended Version)

Details On the Equation $x^{2^l+1}+x+a=0$ over $\mathrm{GF}(2^k)$ (Extended Version) On the Equation $x^{2^l+1}+x+a=0$ over $\mathrm{GF}(2^k)$ (Extended Version)

2008-10-22

arxiv.org/abs/0810.4015v3

Computer Science

Discrete Mathematics

Extended version of the paper with the same title which earlier appeared in Finite Fields and their applications

Scientific paper

In this paper, the polynomials $P_a(x)=x^{2^l+1}+x+a$ with $a\in\mathrm{GF}(2^k)$ are studied. New criteria for the number of zeros of $P_a(x)$ in $\mathrm{GF}(2^k)$ are proved. In particular, a criterion for $P_a(x)$ to have exactly one zero in $\mathrm{GF}(2^k)$ when $\gcd(l,k)=1$ is formulated in terms of the values of permutation polynomials introduced by Dobbertin. We also study the affine polynomial $a^{2^l}x^{2^{2l}}+x^{2^l}+ax+1$ which is closely related to $P_a(x)$. In many cases, explicit expressions for calculating zeros of these polynomials are provided.

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