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 y2f(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

References

  1. Washington, L.C. Elliptic Curves: Number Theory and Cryptography. Chapman & Hall, 2008.
  2. Lenstra Jr, H.W. Factoring integers with elliptic curves. Annals of Mathematics, vol. 126, 1987, 649–673.
  3. Diffie W., M. Hellman. New directions in cryptography. IEEE Transactions on Information Theory, Vol. 22, 1976, 644–654.
  4. Coppersmith, D., A.M. Odlzyko, and R. Schroeppel. Discrete logarithms in GF(p). Algorithmica, Vol. 1, 1986, 1–15.
  5. 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.
  6. Silverman, J.H. The Arithmetic of Elliptic Curves, Springer Verlag, 2009.
  7. 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.
  8. Shankar, B.R. Combinatorial properties of permutation polynomials over some finite rings Zn. IJSDI age, Vol. 1, 1985, 1–6.
  9. Lidl R., H. Niederreiter. Finite fields and their applications. Handbook of Algebra, Vol. 1, 1996, 321–363.
  10. 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.
  11. Gauss, C. F. Disquisitiones Arithmeticae, 1801. English translation by Arthur A. Clarke. Springer-Verlag, New York, 1986.

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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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