# CONSTRUCTING PAIRING-FRIENDLY CURVES WITH VARIABLE CM DISCRIMINANT

• Published : 2012.01.31

#### 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.

#### References

1. 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.
2. 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. https://doi.org/10.1007/s001459900040
3. 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.
4. 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.
5. 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.
6. 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. https://doi.org/10.1006/jsco.1996.0125
7. F. Brezing and A. Weng, Elliptic curves suitable for pairing based cryptography, Des. Codes Cryptogr. 37 (2005), no. 1, 133-141. https://doi.org/10.1007/s10623-004-3808-4
8. 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.
9. D. Freeman, M. Scott, and E. Teske, A taxonomy of pairing-friendly elliptic curves, J. Cryptology 23 (2010), no. 2, 224-280. https://doi.org/10.1007/s00145-009-9048-z
10. S. Galbraith, J. McKee, and P. Valenca, Ordinary abelian varieties having small embedding degree, Finite Fields Appl. 13 (2007), no. 4, 800-814. https://doi.org/10.1016/j.ffa.2007.02.003
11. T. W. Hungerford, Algebra, Graduate Texts in Mathematics, Vol. 73, Springer, Heidelberg, 1996.
12. 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.
13. 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.
14. 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.
15. R. Sakai, K. Ohgishi and M. Kasahara, Cryptosystems based on pairing, The 2000 Symposium on Cryptography and Information Security(SCIS 2000), 2000.
16. J. H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, Berlin, Germany, 1986.
17. A. V. Sutherland, Computing Hilbert class polynomials with the Chinese Remainder Theorem, Math. Comp. 80 (2011), no. 273, 501-538.

#### Cited by

1. Generating pairing-friendly elliptic curve parameters using sparse families vol.12, pp.2, 2018, https://doi.org/10.1515/jmc-2017-0024