Efficient Pairing Computation on Genus 2 Curves in Projective Coordinates

  • Xinxin Fan
  • Guang Gong
  • David Jao
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5381)


In recent years there has been much interest in the development and the fast computation of bilinear pairings due to their practical and myriad applications in cryptography. Well known efficient examples are the Weil and Tate pairings and their variants such as the Eta and Ate pairings on the Jacobians of (hyper-)elliptic curves. In this paper, we consider the use of projective coordinates for pairing computations on genus 2 hyperelliptic curves over prime fields. We generalize Chatterjee et. al.’s idea of encapsulating the computation of the line function with the group operations to genus 2 hyperelliptic curves, and derive new explicit formulae for the group operations in projective and new coordinates in the context of pairing computations. When applying the encapsulated explicit formulae to pairing computations on supersingular genus 2 curves over prime fields, theoretical analysis shows that our algorithm is faster than previously best known algorithms whenever a field inversion is more expensive than about fifteen field multiplications. We also investigate pairing computations on non-supersingular genus 2 curves over prime fields based on the new formulae, and detail the various techniques required for efficient implementation.


Genus 2 hyperelliptic curves Tate pairing Miller’s algorithm Projective coordinates Efficient Implementation 


  1. 1.
    Avanzi, R.M., Cohen, H., Doche, C., Frey, G., Lange, T., Nguyen, K., Vercauteren, F.: Handbook of Elliptic and Hyperelliptic Curve Cryptography. Chapman & Hall/CRC, Boca Raton (2006)zbMATHGoogle Scholar
  2. 2.
    Barreto, P.L.S.M., Galbraith, S., Ó hÉigeartaigh, C., Scott, M.: Efficient Pairing Computation on Supersingular Abelian Varieties. Design, Codes and Cryptography 42, 239–271 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barreto, P.L.S.M., Kim, H.Y., Lynn, B., Scott, M.: Efficient Algorithm for Pairing-Based Cryptosystems. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, p. 354. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    Boneh, D., Franklin, M.: Identity-Based Encryption from the Weil Pairing. SIAM Journal of Computing 32(3), 586–615 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Boneh, D., Lynn, B., Shacham, H.: Short Signatures from the Weil Pairing. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 514–532. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  6. 6.
    Chatterjee, S., Sarkar, P., Barua, R.: Efficient Computation of Tate Pairing in Projective Coordinate over General Characteristic Fields. In: Park, C.-s., Chee, S. (eds.) ICISC 2004. LNCS, vol. 3506, pp. 168–181. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    Choie, Y., Lee, E.: Implementation of Tate Pairing on Hyperelliptic Curve of Genus 2. In: Lim, J.-I., Lee, D.-H. (eds.) ICISC 2003. LNCS, vol. 2971, pp. 97–111. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  8. 8.
    Cocks, C., Pinch, R.G.E.: Identity-based Cryptosystems Based on the Weil Pairing (Unpublished manuscript) (2001)Google Scholar
  9. 9.
    Duursma, I.M., Lee, H.-S.: Tate pairing implementation for hyperelliptic curves y 2 = x p− x + d. In: Laih, C.-S. (ed.) ASIACRYPT 2003. LNCS, vol. 2894, pp. 111–123. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  10. 10.
    Galbraith, S.D., McKee, J.F., Valença, P.C.: Ordinary Abelian Varieties Having Small Embedding Degree. Finite Fields and Their Applications 13(4), 800–814 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Freeman, D.: Constructing Pairing-Friendly Genus 2 Curves over Prime Fields with Ordinary Jacobians. In: Takagi, T., Okamoto, T., Okamoto, E., Okamoto, T. (eds.) Pairing 2007. LNCS, vol. 4575, pp. 152–176. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. 12.
    Frey, G., Lange, T.: Fast Bilinear Maps from The Tate-Lichtenbaum Pairing on Hyperelliptic Curves. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 466–479. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Frey, G., Rück, H.-G.: A Remark Concerning m-Divisibility and the Discrete Logarithm Problem in the Divisor Class Group of Curves. Mathematics of Computation 62(206), 865–874 (1994)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Granger, R., Hess, F., Oyono, R., Thériault, N., Vercauteren, F.: Ate Pairing on Hyperelliptic Curves. In: Naor, M. (ed.) EUROCRYPT 2007. LNCS, vol. 4515, pp. 430–447. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  15. 15.
    Hankerson, D., Menezes, A., Vanstone, S.: Guide to Elliptic Curve Cryptography. Springer, New York (2004)zbMATHGoogle Scholar
  16. 16.
    Ó hÉigeartaigh, C., Scott, M.: Pairing Calculation on Supersingular Genus 2 Curves. In: Biham, E., Youssef, A.M. (eds.) SAC 2006. LNCS, vol. 4356, pp. 302–316. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  17. 17.
    Hess, F., Smart, N.P., Vercauteren, F.: The Eta Pairing Revisited. IEEE Transactions on Information Theory 52(10), 4595–4602 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hitt, L.: Families of Genus 2 Curves with Small Embedding Degree, Cryptology ePrint Archive, Report 2007/001 (2007),
  19. 19.
    Joux, A.: A One-Round Protocol for Tripartite Diffie-Hellman. In: Bosma, W. (ed.) ANTS 2000. LNCS 1838, vol. 1838, pp. 385–394. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  20. 20.
    Karatsuba, A., Ofman, Y.: Multiplication of Multidigit Numbers on Automata. Soviet Physics Doklady (English Translation) 7(7), 595–596 (1963)Google Scholar
  21. 21.
    Kawazoe, M., Takahashi, T.: Pairing-friendly Hyperelliptic Curves of Type y2 = x5 + ax, Cryptology ePrint Archive, Report 2008/026 (2008),
  22. 22.
    Lange, T.: Formulae for Arithmetic on Genus 2 Hyperelliptic Curves. Applicable Algebra in Engineering, Communication and Computing 15(5), 295–328 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lee, E., Lee, H.-S., Lee, Y.: Eta Pairing Computation on General Divisors over Hyperelliptic Curves y 2 = x 7 − x ±1. In: Takagi, T., Okamoto, T., Okamoto, E., Okamoto, T. (eds.) Pairing 2007. LNCS, vol. 4575, pp. 349–366. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  24. 24.
    Menezes, A., Okamoto, T., Vanstone, S.A.: Reducing Elliptic Curve Logarithms to a Finite Field. IEEE Transactions on Information Theory 39(5), 1639–1646 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Menezes, A., van Oorschot, P.C., Vanstone, S.A.: Handbook of Applied Cryptography. Chapman & Hall/CRC, Boca Raton (1997)zbMATHGoogle Scholar
  26. 26.
    Miller, V.S.: Short Programs for Functions on Curves (Unpublished manuscript) (1986),
  27. 27.
    Miyamoto, Y., Doi, H., Matsuo, K., Chao, J., Tsujii, S.: A Fast Addition Algorithm of Genus Two Hyperelliptic Curve. In: The 2002 Symposium on Cryptography and Information Security - SCIS 2002, pp. 497–502 (2002) (in Japanese)Google Scholar
  28. 28.
    Mumford, D.: Tata Lectures on Theta II. In: Prog. Math., vol. 43. Birkhäuser (1984)Google Scholar
  29. 29.
    Sakai, R., Ohgishi, K., Kasahara, M.: Cryptosystems Based on Pairings. In: Proceedings of the 2000 Symposium on Cryptography and Information Security - SCIS 2002, Okinawa, Japan, pp. 26–28 (2000)Google Scholar
  30. 30.
    Scott, M.: MIRACL (Multiprecision Integer and Rational Arithmetic C/C++ Library),
  31. 31.
    Scott, M.: Scaling Security in Pairing-based Protocols, Cryptology ePrint Archive, Report 2005/139 (2005),
  32. 32.
    Solinas, J.: Generalized Mersenne Primes, Centre for Applied Cryptographic Research (CACR) Technical Reports, CORR 99-39,

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Xinxin Fan
    • 1
  • Guang Gong
    • 1
  • David Jao
    • 2
  1. 1.Department of Electrical and Computer EngineeringUniversity of WaterlooWaterlooCanada
  2. 2.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada

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