P. Lisonek, On the connection between Kloosterman sums and elliptic curves. Proceedings of the 5th International Conference on Sequences and Their Applications (SETA 2008) (S. Golomb al., Eds.), Lecture Notes in Computer Science Vol. 5203, pp. 182-187, Springer 2008.We explore the known connection of Kloosterman sums on fields of characteristic 2 and 3 with the number of points on certain elliptic curves over these fields. We use this connection to prove results on the divisibility of Kloosterman sums, and to compute numerical examples of zeros of Kloosterman sums on binary and ternary fields of large orders. We also show that this connection easily yields some formulas due to Carlitz that were recently used to prove certain non-existence results on Kloosterman zeros in subfields.
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