Algorithmic Number Theory

Volume 1838 of the series Lecture Notes in Computer Science pp 395-404

On Exponential Sums and Group Generators for Elliptic Curves over Finite Fields

  • David R. KohelAffiliated withSchool of Mathematics and Statistics, University of Sydney
  • , Igor E. ShparlinskiAffiliated withDepartment of Computing, Macquarie University

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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.