Constructing Elliptic Curve Including Subgroup with Low Hamming Order

Abstract:

In practical applications of pairing-based cryptosystems, the efficiency of pairing computation is a crucial factor. Recently, there have been many improvements for the computation of Tate pairing, which focuses on the arithmetical operations under given elliptic curve. Based to the characteristics that Miller’s algorithm will be improved tremendous if there are subgroups with order of low hamming prime above the elliptic curve, an algorithm of generating primes of low hamming with weight 3 is given in this paper. Then, we present an effective generation method of elliptic curve, which enable it feasible that there is certain some subgroup of low hamming prime order. The improvement of paring computation is marked above the elliptic curve generating by our method.

• Periodical: Advanced Materials Research (Volumes 113-116)
• Pages: 6-9
• DOI: https://doi.org/10.4028/www.scientific.net/AMR.113-116.6
• Online since: June 2010
• Authors: Mao Cai Wang, Han Ping Hu, Guang Ming Dai, Lei Pen
• Keywords: Elliptic Curve, Miller's Algorithm, Pairing Based Cryptosystem, Prime Test

References

[1] Boyen X. A promenade through the new cryptography of bilinear pairings. ITW 2006, IEEE Press, 2006: 19-23.

DOI: 10.1109/itw.2006.1633773

[2] Wang M.C., Hu H.P. and Dai G.M. An identity-based signature scheme for mobile business. ICIC Express Letters, 2010(2): 565-569.

[3] Barreto P., Galbraith S., Eigeartaigh C., etc. Efficient pairing computation on supersingular abelian varieties. Designs, Codes and Cryptography, 2007(42): 239-271.

DOI: 10.1007/s10623-006-9033-6

[4] Barreto P. S. L. M., Kim H. Y., Lynn B., and Scott M. Efficient Algorithms for Pairing-Based Cryptosystems. CRYPTO'2002, Springer-Verlag, 2002: 354-368.

DOI: 10.1007/3-540-45708-9_23

[5] Galbraith S. D., Harrison K. and Soldera D. Implementing the tate pairing. ANTS-V, Springer-Verlag, 2002: 324-337.

[6] Duursma I. and Lee H. Tate pairing implementation for hyperelliptic curves 2 p y x x d= − + . In ASIACRYPTO'2003, Springer-Verlag, 2003: 111-123.

DOI: 10.1007/978-3-540-40061-5_7

[7] Lynn B. ON THE IMPLEMENTATION OF PAIRING-BASED CRYPTOSYSTEMS. Stanford University, Ph. D thesis, 2007. 6.

[8] Stogbauer M. Efficient Algorithms for Pairing-Based Cryptosystems. Darmstadt University of Technology, Ph. D thesis, 2004. 1.

[9] Scott M., Costigan N., and Abdulwahab W. Implementing Cryptographic Pairings on Smartcards. CHES'2006, Springer-Verlag, 2006: 134-147.

DOI: 10.1007/11894063_11

[10] Kwon S. Efficient Tate Pairing Computation for Elliptic Curves over Binary Fields. ACISP'2005, Springer-Verlag, 2005: 134-145.

DOI: 10.1007/11506157_12

[11] Duursma I. and Lee H. Tate pairing implementation for hyperelliptic curves 2 p y x x d= − + . Asiacrypt'2003, Springer-Verlag, 2003: 111-123.

DOI: 10.1007/978-3-540-40061-5_7