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)**

## 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

### References

- Washington, L.C. Elliptic Curves: Number Theory and Cryptography. Chapman & Hall, 2008.
- Lenstra Jr, H.W. Factoring integers with elliptic curves. Annals of Mathematics, vol. 126, 1987, 649–673.
- Diffie W., M. Hellman. New directions in cryptography. IEEE Transactions on Information Theory, Vol. 22, 1976, 644–654.
- Coppersmith, D., A.M. Odlzyko, and R. Schroeppel. Discrete logarithms in
*GF*(*p*). Algorithmica, Vol. 1, 1986, 1–15. - Liu, D., D. Huang, P. Luo, Y. Dai. New schemes for sharing points on an elliptic curve. Computers & Mathematics with Applications, Vol. 56, 2008, 1556–1561.
- Silverman, J.H. The Arithmetic of Elliptic Curves, Springer Verlag, 2009.
- Lidl, R. On cryptosystems based on polynomials and finite fields. Advances in Cryptology: Proceedings of EUROCRYPT 84-A Workshop on the Theory and Application of Cryptographic Techniques, Paris, France, April 1984, 1985, p. 10.
- Shankar, B.R. Combinatorial properties of permutation polynomials over some finite rings
*Z*. IJSDI age, Vol. 1, 1985, 1–6._{n} - Lidl R., H. Niederreiter. Finite fields and their applications. Handbook of Algebra, Vol. 1, 1996, 321–363.
- Chen, Y.L., J. Ryu, O.Y. Takeshita. A simple coefficient test for cubic permutation polynomials over integer rings. Communications Letters, IEEE, Vol. 10, 2006, 549–551.
- Gauss, C. F. Disquisitiones Arithmeticae, 1801. English translation by Arthur A. Clarke. Springer-Verlag, New York, 1986.

## Related papers

## Cite this paper

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