Finite field elements of high order arising from modular curves
- 67 Downloads
In this paper, we recursively construct explicit elements of provably high order in finite fields. We do this using the recursive formulas developed by Elkies to describe explicit modular towers. In particular, we give two explicit constructions based on two examples of his formulas and demonstrate that the resulting elements have high order. Between the two constructions, we are able to generate high order elements in every characteristic. Despite the use of the modular recursions of Elkies, our methods are quite elementary and require no knowledge of modular curves. We compare our results to a recent result of Voloch. In order to do this, we state and prove a slightly more refined version of a special case of his result.
KeywordsFinite field Modular curves
Mathematics Subject Classification (2000)11T71 11T55
Unable to display preview. Download preview PDF.
- 4.Conflitti A.: On elements of high order in finite fields. In: Cryptography and Computational Number Theory Singapore, 1999. Progr. Comput. Sci. Appl. Logic, vol. 20, pp. 11–14. Birkhäuser, Basel (2001).Google Scholar
- 5.Elkies N.D.: Explicit modular towers. In: Proceedings of the Thirty-Fifth Annual Allerton Conference on Communication, Control and Computing. Univ. of Illinois at Urbana-Champaign (1998).Google Scholar
- 13.von zur Gathen J., Shparlinski I.: Orders of Gauss periods in finite fields. In: Algorithms and Computations, Cairns, 1995. Lecture Notes in Comput. Sci., vol. 1004, pp. 208–215. Springer, Berlin (1995) (Also appeared as Orders of Gauss periods in finite fields. Applicable Algebra in Engineering, Communication and Computing, 9, 1998, 15–24).Google Scholar
- 14.von zur Gathen J., Shparlinski I.: Gauß periods in finite fields. In: Finite Fields and Applications, Augsburg, 1999, pp. 162–177. Springer, Berlin (2001).Google Scholar