[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/BKMS.2012.49.1.075

CONSTRUCTING PAIRING-FRIENDLY CURVES WITH VARIABLE CM DISCRIMINANT

Lee, Hyang-Sook (Department of Mathematics Ewha Womans University)
Park, Cheol-Min (Institute of Mathematical Sciences Ewha Womans University)

Publication Information

Bulletin of the Korean Mathematical Society / v.49, no.1, 2012 , pp. 75-88 More about this Journal

Abstract

A new algorithm is proposed for the construction of Brezing-Weng-like elliptic curves such that polynomials defining the CM discriminant are linear. Using this construction, new families of curves with variable discriminants and embedding degrees of $k{\in}\{8,16,20,24\}$ , which were not covered by Freeman, Scott, and Teske [9], are presented. Our result is useful for constructing elliptic curves with larger and more flexible discriminants.

Keywords

elliptic curves; pairing-friendly curves; CM discriminant;

Citations & Related Records

(Related Records In Web of Science)
Times Cited By SCOPUS : 0

Reference

1	A. V. Sutherland, Computing Hilbert class polynomials with the Chinese Remainder Theorem, Math. Comp. 80 (2011), no. 273, 501-538.
2	W. Bosma, J. Cannon, and C. Playoust. The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235-265. DOI ScienceOn
3	F. Brezing and A. Weng, Elliptic curves suitable for pairing based cryptography, Des. Codes Cryptogr. 37 (2005), no. 1, 133-141. DOI ScienceOn
4	D. Freeman, Constructing Pairing-Friendly Elliptic Curves with Embedding Degree 10,Algorithmic Number Theory Symposium ANTS-VII, Lecture Notes in Computer Science, Vol. 4076, pp. 452-465, Springer-Verlag, 2006.
5	D. Freeman, M. Scott, and E. Teske, A taxonomy of pairing-friendly elliptic curves, J. Cryptology 23 (2010), no. 2, 224-280. DOI ScienceOn
6	S. Galbraith, J. McKee, and P. Valenca, Ordinary abelian varieties having small embedding degree, Finite Fields Appl. 13 (2007), no. 4, 800-814. DOI ScienceOn
7	T. W. Hungerford, Algebra, Graduate Texts in Mathematics, Vol. 73, Springer, Heidelberg, 1996.
8	A. Joux, A one round protocol for tripartite Diffie-Hellman, Proceedings of Algorithmic Number Theory Symposium, ANTS-IV, Lecture Notes in Computer Science, Vol. 1838, pp. 385-394, Springer-Verlag, 2000.
9	E. Kachisa, E. Schaefer, and M. Scott, Constructing Brezing-Weng pairing-friendly elliptic curves using elements in the cyclotomic field, Pairing-based cryptography-Pairing 2008, 126-135, Lecture Notes in Comput. Sci., 5209, Springer, Berlin, 2008.
10	H.-S. Lee and C.-M. Park, Generating pairing-friendly curves with the CM equation of degree 1, Pairing 2009, vol. 5671, Lecture Notes in Computer Science, page 66-77, Springer-Verlag, 2009.
11	R. Sakai, K. Ohgishi and M. Kasahara, Cryptosystems based on pairing, The 2000 Symposium on Cryptography and Information Security(SCIS 2000), 2000.
12	J. H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, Berlin, Germany, 1986.
13	D. Boneh and M. Franklin, Identity-based encryption from the Weil pairing. Advances in Cryptography, Proceedings of Crypto 2001, Lecture Notes in Computer Science, Vol. 2139, pp. 213-229, Springer-Verlag, 2001.
14	R. M. Avanzi, H. Cohen, C. Doche, G. Frey, T. Lange, K. Nguyen, and F. Vercauteren, Handbook of Elliptic and Hyperelliptic Curve Cryptography, Chapman & Hall/CRC, Sydney, 2006.
15	R. Balasubramanian and N. Koblitz, The improbability that an elliptic curve has subex-ponential discrete log problem under the Menezes-Okamoto-Vanstone algorithm, J. Cryptology 11 (1998), no. 2, 141-145. DOI ScienceOn
16	P. S. L. M. Barreto and M. Naehrig, Pairing-friendly elliptic curves of prime order, Proceedings of SAC 2005-Workshop on Selected Areas in Cryptography, Lecture Notes in Computer Science, vol. 3897, pp 319-331, Springer-Verlag, 2006.
17	D. Boneh, B. Lynn, and H. Shacham, Short signatures from the Weil pairing, Advances in Cryptology: Proceedings of Asiacrypt 2001, Lecture Notes in Computer Science, Vol. 2248, pp. 514-532, Springer-Verlag, 2002.