Abstract
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 field. The bound is applied to prove the existence of group generators for the set of points on an elliptic curve over \(\mathbb{F}_{q}\) among certain sets of bounded size. We apply this estimate to obtain a deterministic O(q 1/2 + ε) algorithm for finding generators of the group in echelon form, and in particular to determine its group structure.
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Kohel, D.R., Shparlinski, I.E. (2000). On Exponential Sums and Group Generators for Elliptic Curves over Finite Fields. In: Bosma, W. (eds) Algorithmic Number Theory. ANTS 2000. Lecture Notes in Computer Science, vol 1838. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10722028_24
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DOI: https://doi.org/10.1007/10722028_24
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