Abstract
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.
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Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24(3–4), 235–265 (1997) (Computational algebra and number theory, London, 1993)
Cheng Q.: the construction of finite field elements of large order. Finite Fields Appl. 11(3), 358–366 (2005)
Cheng Q.: finite field extensions with large order elements. SIAM J. Discrete Math 21(3), 726–730 (2007)
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).
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).
Gao S.: Elements of provable high orders in finite fields. Proc. Amer. Math. Soc. 127(6), 1615–1623 (1999)
Gao S., Vanstone S.A.: On orders of optimal normal basis generators. Math. Comp. 64(211), 1227–1233 (1995)
Gao S., von zur Gathen J., Panario D.: Gauss periods: orders and cryptographical applications. Math. Comp 67(221), 343–352 (1998) (With microfiche supplement)
Garcia A., Stichtenoth H.: A tower of Artin-Schreier extensions of function fields attaining the Drinfel′ d-Vlăduţ bound. Invent. Math. 121(1), 211–222 (1995)
Garcia A., Stichtenoth H.: Asymptotically good towers of function fields over finite fields. C. R. Acad. Sci. Paris Sér. I Math. 322(11), 1067–1070 (1996)
Ireland K., Rosen M.: A Classical Introduction to Modern Number Theory, 2nd edn. Springer-Verlag, New York (1990)
Voloch J.F.: On the order of points on curves over finite fields. Integers. 7(A49), 4 (2007)
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).
von zur Gathen J., Shparlinski I.: Gauß periods in finite fields. In: Finite Fields and Applications, Augsburg, 1999, pp. 162–177. Springer, Berlin (2001).
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Communicated by J.D. Key.
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Burkhart, J.F., Calkin, N.J., Gao, S. et al. Finite field elements of high order arising from modular curves. Des. Codes Cryptogr. 51, 301–314 (2009). https://doi.org/10.1007/s10623-008-9262-y
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DOI: https://doi.org/10.1007/s10623-008-9262-y