On the solutions of an elliptic curve over a field of characteristic two

Elliptic curves lie at the intersection of coding and cryptography. A key problem for both subjects is the determination of the number of points on the curve. For coding theory, the number of points one obtains is approximately the length of the code. For cryptography it is important that the group of points contain as large a subgroup of prime order as possible. In both cases the exact number of points is required. Using character theory and results on Kloosterman sums, information is obtained on the structure of solutions of a nonsingular elliptic curve over a finite field of characteristic two. A related enumeration problem of certain irreducible polynomials is also considered