Counting The Number Of Points On Affine Diagonal Curves

The number of points on affine diagonal curves aX(m) + bY(n) = c over finite fields can be computed in terms of cyclotomic numbers. The approach of Berndt, Evans and Williams [1] is to express the number of points in terms of generalized Jacobi sums, then to relate the Jacobi sums J(r)(chi (u), chi (v)) to cyclotomic numbers. In this article we present the direct elementary method for the number of points on the affine curves aX(m) + bY(n) = c over finite fields in terms of cyclotomic numbers. This approach is applicable when explicit formulas are already known for cyclotomic numbers, and circumvents the use of Jacobi sums. It generalizes to the determination of the number of points on affine diagonal hypersurfaces of higher dimension. The curves for which this method applies includes examples of elliptic and hyperelliptic curves which are of interest for public-key cryptosystems, coding theory and the design and analysis of sequences.