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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.
- Elliptic curves
- Permutation polynomials
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Cite this paperAPA
Meemark, Y., & Wongpradit, A. (2011). Permutation polynomials and elliptic curves, Notes on Number Theory and Discrete Mathematics, 17(4), 1-8.Chicago
Meemark, Yotsanan, and Attawut Wongpradit. “Permutation Polynomials and Elliptic Curves.” Notes on Number Theory and Discrete Mathematics 17, no. 4 (2011): 1-8.MLA
Meemark, Yotsanan, and Attawut Wongpradit. “Permutation Polynomials and Elliptic Curves.” Notes on Number Theory and Discrete Mathematics 17.4 (2011): 1-8. Print.