Yotsanan Meemark and Attawut Wongpradit

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 17, 2011, Number 4, Pages 1—8

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## Details

### 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* : *y*^{2} = *f*(*x*), where *f*(*x*) is a cubic permutation polynomial over some finite commutative ring *R*. In case *R* is the finite field *F _{q}*, 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*=

*Z*, the ring of integers modulo a square-free

_{n}*n*. In addition, we introduce a shift-invariant elliptic curve which is an elliptic curve

*E*:

*y*

^{2}=

*f*(

*x*), where

*y*

^{2}−

*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

*F*and

_{q}*Z*.

_{n}### 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.