On the Connection between Kloosterman Sums and Elliptic Curves
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.
KeywordsKloosterman sum elliptic curve finite field
Unable to display preview. Download preview PDF.
- 2.Carlitz, C.: Kloosterman Sums and Finite Field Extensions. Acta Arithmetica XVI, 179–193 (1969)Google Scholar
- 3.Charpin, P., Gong, G.: Hyperbent Functions, Kloosterman Sums and Dickson Polynomials. IEEE Trans. Inform. Theory (to appear)Google Scholar
- 5.Enge, A.: Elliptic Curves and Their Applications to Cryptography: An Introduction. Kluwer Academic Publishers, Boston (1999)Google Scholar
- 6.Garaschuk, K., Lisoněk, P.: On Ternary Kloosterman Sums modulo 12. Finite Fields Appl. (to appear)Google Scholar
- 11.Lercier, R., Lubicz, D., Vercauteren, F.: Point Counting on Elliptic and Hyperelliptic Curves. In: Cohen, H., Frey, G. (eds.) Handbook of Elliptic and Hyperelliptic Curve Cryptography. Chapman & Hall/CRC, Boca Raton (2006)Google Scholar
- 13.Moisio, M.: Kloosterman Sums, Elliptic Curves, and Irreducible Polynomials with Prescribed Trace and Norm. Acta Arithmetica (to appear)Google Scholar