Advertisement

Faster Squaring in the Cyclotomic Subgroup of Sixth Degree Extensions

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6056)

Abstract

This paper describes an extremely efficient squaring operation in the so-called ‘cyclotomic subgroup’ of \(\mathbb{F}_{q^6}^{\times}\), for \(q \equiv 1 \bmod{6}\). Our result arises from considering the Weil restriction of scalars of this group from \(\mathbb{F}_{q^6}\) to \(\mathbb{F}_{q^2}\), and provides efficiency improvements for both pairing-based and torus-based cryptographic protocols. In particular we argue that such fields are ideally suited for the latter when the field characteristic satisfies \(p \equiv 1 \pmod{6}\), and since torus-based techniques can be applied to the former, we present a compelling argument for the adoption of a single approach to efficient field arithmetic for pairing-based cryptography.

Keywords

Pairing-based cryptography torus-based cryptography finite field arithmetic 

References

  1. 1.
    Bailey, D.V., Paar, C.: Optimal Extension Fields for Fast Arithmetic in Public-Key Algorithms. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 472–485. Springer, Heidelberg (1998)Google Scholar
  2. 2.
    Barreto, P., Galbraith, S.D., ÓhÉigeartaigh, C., Scott, M.: Efficient Pairing Computation on Supersingular Abelian Varieties. Designs, Codes and Cryptography 42(3), 239–271 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Barreto, P., Kim, H., Lynn, B., Scott, M.: Efficient Algorithms for Pairing-Based Cryptosystems. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 354–368. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    Barreto, P., Naehrig, M.: Pairing-Friendly Elliptic Curves of Prime Order. In: Preneel, B., Tavares, S. (eds.) SAC 2005. LNCS, vol. 3897, pp. 319–331. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    Benger, N., Scott, M.: Constructing Tower Extensions for the implementation of Pairing-Based Cryptography (Preprint)Google Scholar
  6. 6.
    Blake, I.F., Seroussi, G., Smart, N.P.: Advances in Elliptic Curves in Cryptography. Cambridge University Press, Cambridge (2005)CrossRefGoogle Scholar
  7. 7.
    Boneh, D., Boyen, X., Shacham, H.: Short Group Signatures. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 41–55. Springer, Heidelberg (2004)Google Scholar
  8. 8.
    Chung, J., Hasan, M.A.: Asymmetric Squaring Formulae. In: IEEE Symposium on Computer Arithmetic, pp. 113–122 (2007)Google Scholar
  9. 9.
    Devegili, A.J., ÓhÉigeartaigh, C., Scott, M., Dahab, R.: Multiplication and Squaring on Pairing-Friendly Fields, http://eprint.iacr.org/2006/471
  10. 10.
    Devegili, A.J., Scott, M., Dahab, R.: Implementing Cryptographic Pairings over Barreto-Naehrig Curves. In: Takagi, T., Okamoto, T., Okamoto, E., Okamoto, T. (eds.) Pairing 2007. LNCS, vol. 4575, pp. 197–207. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    van Dijk, M., Granger, R., Page, D., Rubin, K., Silverberg, A., Stam, M., Woodruff, D.: Practical cryptography in high dimensional tori. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 234–250. Springer, Heidelberg (2005)Google Scholar
  12. 12.
    Galbraith, S.D., Harrison, K., Soldera, D.: Implementing the Tate pairing. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 324–337. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  13. 13.
    Galbraith, S.D., Scott, M.: Exponentiation in Pairing-Friendly Groups Using Homomorphisms. In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 211–224. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    Gallant, R., Lambert, J., Vanstone, S.: Faster Point Multiplication on Elliptic Curves with Efficient Endomorphisms. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 190–200. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  15. 15.
    Granger, R., Page, D., Smart, N.P.: High Security Pairing-Based Cryptography Revisited. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 480–494. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  16. 16.
    Granger, R., Page, D., Stam, M.: A Comparison of CEILIDH and XTR. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 235–249. Springer, Heidelberg (2004)Google Scholar
  17. 17.
    Granger, R., Page, D., Stam, M.: On Small Characteristic Algebraic Tori in Pairing-based Cryptography. LMS Journal of Computation and Mathematics 9, 64–85 (2006)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Hess, F., Vercauteren, F., Smart, N.P.: The Eta Pairing Revisited. IEEE Transactions on Information Theory 52(10), 4595–4602 (2006)CrossRefMathSciNetGoogle Scholar
  19. 19.
    IEEE Draft Standard for Identity-based Public-key Cryptography using Pairings, P1636.3/D1 (2008), http://grouper.ieee.org/groups/1363/IBC/material/P1363.3-D1-200805.pdf
  20. 20.
    IEEE Draft Standard for identity-based cryptographic techniques using pairings, P1363.3/D3 (2009), http://grouper.ieee.org/groups/1363/IBC/index.html
  21. 21.
    Joux, A.: A One Round Protocol for Tripartite Diffie-Hellman. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 385–394. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  22. 22.
    Kachisa, E.J., Schaefer, E.F., Scott, M.: Constructing Brezing-Weng Pairing-Friendly Elliptic Curves Using Elements in the Cyclotomic Field. In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 126–135. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  23. 23.
    Karatsuba, A., Ofman, Y.: Multiplication of Many-Digital Numbers by Automatic Computers. Soviet Physics Doklady 7, 595–596 (1963)Google Scholar
  24. 24.
    Koblitz, N., Menezes, A.J.: Pairing-Based Cryptography at High Security Levels. In: Smart, N.P. (ed.) Cryptography and Coding 2005. LNCS, vol. 3796, pp. 13–36. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  25. 25.
    Lee, E., Lee, H.S., Park, C.M.: Efficient and Generalized Pairing Computation on Abelian Varieties. IEEE Transactions on Information Theory 55(4), 1793–1803 (2009)CrossRefGoogle Scholar
  26. 26.
    Lenstra, A.K., Verheul, E.: The XTR Public Key System. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, pp. 1–19. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  27. 27.
    Lim, S., Kim, S., Yie, I., Kim, J., Lee, H.: XTR extended to GF(p\(^{\mbox{6m}}\)). In: Vaudenay, S., Youssef, A.M. (eds.) SAC 2001. LNCS, vol. 2259, pp. 301–312. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  28. 28.
    Miyaji, A., Nakabayashi, M., Takano, S.: New explicit conditions of elliptic curve traces for FR-reduction. IEICE Trans. Fundamentals E84-A (5), 1234–1243 (2001)Google Scholar
  29. 29.
    Naehrig, M., Barreto, P.S.L.M., Schwabe, P.: On Compressible Pairings and their Computation. In: Vaudenay, S. (ed.) AFRICACRYPT 2008. LNCS, vol. 5023, pp. 371–388. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  30. 30.
    Rubin, K., Silverberg, A.: Torus-Based Cryptography. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 349–365. Springer, Heidelberg (2003)Google Scholar
  31. 31.
    Scott, M., Barreto, P.: Compressed Pairings. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 140–156. Springer, Heidelberg (2004)Google Scholar
  32. 32.
    Scott, M., Benger, N., Charlemagne, M., Perez, L.J.D., Kachisa, E.J.: On the Final Exponentiation for Calculating Pairings on Ordinary Elliptic Curves. In: Shacham, H. (ed.) Pairing 2009. LNCS, vol. 5671, pp. 78–88. Springer, Heidelberg (2009)Google Scholar
  33. 33.
    Smith, P., Skinner, C.: A public-key cryptosystem and a digital signature system based on the Lucas function analogue to discrete logarithms. In: Safavi-Naini, R., Pieprzyk, J.P. (eds.) ASIACRYPT 1994. LNCS, vol. 917, pp. 357–364. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  34. 34.
    Stam, M., Lenstra, A.K.: Efficient Subgroup Exponentiation in Quadratic and Sixth Degree Extensions. In: Kaliski Jr., B.S., Koç, Ç.K., Paar, C. (eds.) CHES 2002. LNCS, vol. 2523, pp. 318–332. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  35. 35.
    Stam, M., Lenstra, A.K.: Speeding Up XTR. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 125–143. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  36. 36.
    Toom, A.L.: The Complexity of a Scheme of Functional Elements realizing the Multiplication of Integers. Soviet Mathematics 4(3), 714–716 (1963)Google Scholar
  37. 37.
    Weil, A.: Adeles and algebraic groups. Progress in Mathematics, vol. 23. Birkhäuser, Boston (1982)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Claude Shannon Institute School of ComputingDublin City UniversityGlasnevin, Dublin 9Ireland

Personalised recommendations