Designs, Codes and Cryptography

, Volume 51, Issue 3, pp 301–314

# Finite field elements of high order arising from modular curves

• Jessica F. Burkhart
• Neil J. Calkin
• Shuhong Gao
• Justine C. Hyde-Volpe
• Kevin James
• Hiren Maharaj
• Shelly Manber
• Jared Ruiz
• Ethan Smith
Article

## 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.

## Keywords

Finite field Modular curves

11T71 11T55

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## Authors and Affiliations

• Jessica F. Burkhart
• 1
• Neil J. Calkin
• 1
• Shuhong Gao
• 1
• Justine C. Hyde-Volpe
• 1
• Kevin James
• 1
• Hiren Maharaj
• 1
• Shelly Manber
• 2
• Jared Ruiz
• 3
• Ethan Smith
• 1
1. 1.Department of Mathematical SciencesClemson UniversityClemsonUSA
2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
3. 3.Department of Mathematics and StatisticsYoungstown State UniversityYoungstownUSA