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

Abstract

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… (More)

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