Finite field elements of high order arising from modular curves

Authors:
Jessica F. Burkhart;Neil J. Calkin;Shuhong Gao;Justine C. Hyde-Volpe;Kevin James;Hiren Maharaj;Shelly Manber;Jared Ruiz;Ethan Smith
Affiliations:
Department of Mathematical Sciences, Clemson University, Clemson, USA 29634-0975;Department of Mathematical Sciences, Clemson University, Clemson, USA 29634-0975;Department of Mathematical Sciences, Clemson University, Clemson, USA 29634-0975;Department of Mathematical Sciences, Clemson University, Clemson, USA 29634-0975;Department of Mathematical Sciences, Clemson University, Clemson, USA 29634-0975;Department of Mathematical Sciences, Clemson University, Clemson, USA 29634-0975;Department of Mathematics, Massachusetts Institute of Technology, Cambridge, USA 02139;Department of Mathematics and Statistics, Youngstown State University, Youngstown, USA 44555;Department of Mathematical Sciences, Clemson University, Clemson, USA 29634-0975
Venue:
Designs, Codes and Cryptography
Year:
2009

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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.