Finite field elements of high order arising from modular curves

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.

[1]  Qi Cheng.  On the construction of finite field elements of large order , 2005, Finite Fields Their Appl..

[2]  H. Stichtenoth,et al.  Asymptotically good towers of function fields over finite fields , 1996 .

[3]  Noam D. Elkies,et al.  Explicit Modular Towers , 2001, math/0103107.

[4]  Wieb Bosma,et al.  Computational Algebra and Number Theory , 1995 .

[5]  Michael Rosen,et al.  A classical introduction to modern number theory , 1982, Graduate texts in mathematics.

[6]  José Felipe Voloch.  On the order of points on curves over finite fields. , 2007 .

[7]  Qi Cheng.  Constructing finite field extensions with large order elements , 2004, SODA '04.

[8]  Igor Shparlinski,et al.  Orders of Gauss Periods in Finite Fields , 1998, Applicable Algebra in Engineering, Communication and Computing.

[9]  Alessandro Conflitti.  On Elements of High Order in Finite Fields , 2001 .

[10]  J. Gathen,et al.  Gauß Periods in Finite Fields , 2001 .

[11]  Joachim von zur Gathen,et al.  Gauss periods: orders and cryptographical applications , 1998, Math. Comput..

[12]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

[13]  Shuhong Gao,et al.  Elements of provable high orders in finite fields , 1999 .

[14]  Igor E. Shparlinski,et al.  Orders of Gauss Periods in Finite Fields , 1995, ISAAC.

[15]  H. Stichtenoth,et al.  A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound , 1995 .

[16]  Shuhong Gao,et al.  On orders of optimal normal basis generators , 1995 .