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

In this paper, the polynomials Pa(x) = x 2l+1 + x + a with a ∈ GF(2) are studied. New criteria for the number of zeros of Pa(x) in GF(2 ) are proved. In particular, a criterion for Pa(x) to have exactly one zero in GF(2 ) 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 l x 2l +x l + ax+ 1 which is closely related to Pa(x). In many cases, explicit expressions for calculating zeros of these polynomials are provided.