On exponential sums and group generators for elliptic curves over finite fields

In the paper an upper bound is established for certain exponential sums, analogous to Gaussian sums, defined on the points of an elliptic curve over a prime finite held. The bound is applied to prove the existence of group generators for the set of points on an elliptic curve over F-q among certain sets of bounded size. We apply this estimate to obtain a deterministic O(q(1/2+epsilon)) algorithm for finding generators of the group in echelon form, and in particular to determine its group structure.