Permutation polynomials and elliptic curves

Yotsanan Meemark and Attawut Wongpradit
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 17, 2011, Number 4, Pages 1–8
Full paper (PDF, 188 Kb)

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Authors and affiliations

Yotsanan Meemark
Department of Mathematics, Faculty of Science, Chulalongkorn University
Bangkok, 10330, Thailand

Attawut Wongpradit
Department of Mathematics, Faculty of Science, Chulalongkorn University
Bangkok, 10330, Thailand

Abstract

In this work, we study the elliptic curve E : y2 = f(x), where f(x) is a cubic permutation polynomial over some finite commutative ring R. In case R is the finite field Fq, it turns out that the group of rational points on E is cyclic of order q +1. This group is a product of cyclic groups if R = Zn, the ring of integers modulo a square-free n. In addition, we introduce a shift-invariant elliptic curve which is an elliptic curve E : y2 = f(x), where y2 − f(x) is a weak permutation polynomial. We end our paper with a necessary and sufficient condition for the existence of a shift-invariant elliptic curve over Fq and Zn.

Keywords

  • Elliptic curves
  • Permutation polynomials

AMS Classification

  • 05A05
  • 11G20

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Cite this paper

Meemark, Y., & Wongpradit, A. (2011). Permutation polynomials and elliptic curves. Notes on Number Theory and Discrete Mathematics, 17(4), 1-8.

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