Bulletin of the Australian Mathematical Society

Table of Contents - June 2010 - Volume 81, Issue 03  

Research Article

ELEMENTS OF HIGH ORDER ON FINITE FIELDS FROM ELLIPTIC CURVES

JOSÉ FELIPE VOLOCHa1

a1 Department of Mathematics, University of Texas, Austin, TX 78712, USA (email: voloch@math.utexas.edu)

Abstract

We discuss the problem of constructing elements of multiplicative high order in finite fields of large degree over their prime field. We obtain such elements by evaluating rational functions on elliptic curves, at points whose order is small with respect to their degree. We discuss several special cases, including an old construction of Wiedemann, giving the first nontrivial estimate for the order of the elements in this construction.

(Received June 03 2009)

2000 Mathematics subject classification

  • primary 14G15; secondary 11G20;
  • 11T06

Keywords and phrases

  • finite fields;
  • elliptic curves;
  • multiplicative group

Footnotes

Research supported by NSA grant MDA904-H98230-09-1-0070.