Efficient Implementations of Index Calculation Methods of Elliptic Curves using Weil`s Theorem

- Journal title : The Journal of the Korea institute of electronic communication sciences
- Volume 11, Issue 7, 2016, pp.693-700
- Publisher : The Korea Institute of Electronic Communication Science
- DOI : 10.13067/JKIECS.2016.11.7.693

Title & Authors

Efficient Implementations of Index Calculation Methods of Elliptic Curves using Weil`s Theorem

Kim, Yong-Tae;

Kim, Yong-Tae;

Abstract

It is important that we can calculate the order of non-supersingular elliptic curves with large prime factors over the finite field GF(q) to guarantee the security of public key cryptosystems based on discrete logarithm problem(DLP). Schoof algorithm, however, which is used to calculate the order of the non-supersingular elliptic curves currently is so complicated that many papers are appeared recently to update the algorithm. To avoid Schoof algorithm, in this paper, we propose an algorithm to calculate orders of elliptic curves over finite composite fields of the forms $GF(2^m)

Keywords

Order Of The Elliptic Curve;Non-Supersingular Elliptic Curve;Schoof Algorithm;Weil`S Theorem;

Language

Korean

References

1.

A. Menezes, Elliptic Curve Public Key Cryptosystems. Kluber Academic Publishers, Dordrecht, 1993.

2.

H. Hasse, "Zur Theorie der abstrakten elliptischen Funktionenkorper, I,II&III," Crelle 174, 1936, pp. 173-177.

3.

R. Schoof, "Elliptic curves over finite fields and the computation of square roots mod p," Mathematics of Computation, vol. 44, no. 170, 1985, pp. 483-494.

4.

N. Koblitz, "Constructing elliptic curve cryptosystems in characteristic 2," Advances in Cryptography-CRYPTO'90, Proc., LNCS 537, Springer, Santa Barbara, USA, Aug., 1991, pp. 156-167.

5.

U. Choi and S. Cho, "Design of Binary Sequence with optimal Cross-correlation Values," J. of the Korea Institute of Electronic Communication Sciences, vol. 6, no. 4, 2011, pp. 539-544.

6.

S. Cho, J. Kim, U. Choi, and S. Kim, "Cross-correlation of linear and nonlinear GMW-sequences generated by the same primitive polynomial on GF($2^p$ )," The Korea Institute of Electronic Communication Sciences 2011 Spring Conf. Busan, Korea, vol. 5, no. 1, June 2011, pp. 155-158.

7.

T. Satoh, "On p-adic point counting algorithms for elliptic curves over finite fields," Algorithmic Number Theory, 5th Int. Symp., ANTS-V, Lecture Notes on Computer Science 2369, Springer, Berlin, July, 2002, pp. 43-66.

8.

J. von zur Gathen and J. Garhard, Modern Computer Algebra. 3rd Ed., Cambridge University Press, Cambridge, 2013.

9.

L. C. Washington, "Elliptic Curves; Number Theory and Cryptography," New York: Chapman&Hall/CRC, 2003.

10.

S. Wolfram, Mathematica. 4th Ed., Wolfram Champaign Research, Inc., New York, 1999.

11.

H. Kim, S. Cho, M. Kwon, and H. An, "A study on the cross sequences," J. of the Korea Institute of Electronic Communication Sciences, vol. 7, no. 1, 2012, pp. 61-67.