Skip to main content
SpringerLink
Log in
Book cover

International Workshop on Security

IWSEC 2012: Advances in Information and Computer Security pp 19–36Cite as

Application of Scalar Multiplication of Edwards Curves to Pairing-Based Cryptography

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 7631))

Abstract

Edwards curves have efficient scalar multiplication algorithms, and their application to pairing-based cryptography has been studied. In particular, if a pairing-friendly curve used in a pairing-based protocol is isomorphic to an Edwards curve, all the scalar multiplication appearing in the protocol can be computed efficiently. In this paper, we extend this idea to pairing-friendly curves not isomorphic but isogenous to Edwards curves, and add to pairing-friendly curves to which Edwards curves can be applied. Above all, pairing-friendly curves with smaller ρ-values provide more efficient pairing computation. Therefore, we investigate whether pairing-friendly curves with the minimal ρ-values are isogenous to Edwards curves for embedding degree up to 50. Based on the investigation, we present parameters of pairing-friendly curves with 160-bit and 256-bit security level at embedding degree 16 and 24, respectively. These curves have the minimal ρ-values and are not isomorphic but isogenous to Edwards curves, and thus our proposed method is effective for these curves.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   72.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aréne, C., Lange, T., Naehrig, M., Ritzenthaler, C.: Faster Pairing Computation of the Tate Pairing. Journal of Number Theory 131, 842–847 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Atkin, A.O.L., Morain, F.: Elliptic Curves and Primarity Proving. Math. Comp. 61(203), 29–68 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bach, E., Shallit, J.: Algorithmic number theory. Efficient algorithms. Foundations of Computing Series, vol. 1. MIT Press, Cambridge (1996)

    Google Scholar 

  4. Balasubramanian, R., Koblitz, N.: The Improbability that an Elliptic Curve has Subexponential Discrete Log Problem under the Menezes-Okamoto-Vanstone Algorithm. J. Cryptology 11(2), 141–145 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barreto, P.S.L.M., Galbraith, S., O’hEigeartaigh, C., Scott, M.: Efficient Pairing Computation on Supersingular Abelian Varieties. Designs, Codes and Cryptography, 239–271 (2004)

    Google Scholar 

  6. Barreto, P.S.L.M., Lynn, B., Scott, M.: Constructing Elliptic Curves with Prescribed Embedding Degrees. In: Cimato, S., Galdi, C., Persiano, G. (eds.) SCN 2002. LNCS, vol. 2576, pp. 257–267. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  7. Barreto, P.S.L.M., 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)

    Chapter  Google Scholar 

  8. Benger, N., Scott, M.: Constructing Tower Extensions of Finite Fields for Implementation of Pairing-Based Cryptography. In: Hasan, M.A., Helleseth, T. (eds.) WAIFI 2010. LNCS, vol. 6087, pp. 180–195. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  9. Bernstein, D.J., Birkner, P., Joye, M., Lange, T., Peters, C.: Twisted Edwards Curves. In: Vaudenay, S. (ed.) AFRICACRYPT 2008. LNCS, vol. 5023, pp. 389–405. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  10. Bernstein, D.J., Lange, T.: Faster Addition and Doubling on Elliptic Curves. In: Kurosawa, K. (ed.) ASIACRYPT 2007. LNCS, vol. 4833, pp. 29–50. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  11. Boneh, D., Boyen, X.: Efficient Selective-ID Secure Identity-Based Encryption Without Random Oracles. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 223–238. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  12. Boneh, D., Franklin, M.: Identity-Based Encryption from the Weil Pairing. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 213–229. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  13. Brezing, F., Weng, A.: Elliptic Curves Suitable for Pairing based Cryptography. Designs, Codes and Cryptography 37, 133–141 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cocks, C., Pinch, R.G.E.: Identity-based Cryptosystems based on the Weil pairing. Unpublished manuscript (2001)

    Google Scholar 

  15. Cohen, H., Miyaji, A., Ono, T.: Efficient Elliptic Curve Exponentiation Using Mixed Coordinates. In: Ohta, K., Pei, D. (eds.) ASIACRYPT 1998. LNCS, vol. 1514, pp. 51–65. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  16. Das, M.P.L., Sarkar, P.: Pairing Computation on Twisted Edwards Form Elliptic Curves. In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 192–210. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  17. Dupont, P., Enge, A., Morain, F.: Building Curves with Arbitrary Small MOV Degree over Finite Prime Fields. Journal of Cryptology 18, 79–89 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Edwards, H.M.: A Normal Form for Elliptic Curves. Bulletin of the American Mathematical Society 44, 393–422 (2007)

    Article  MATH  Google Scholar 

  19. Freeman, D.: Constructing Pairing-Friendly Elliptic Curves with Embedding Degree 10. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 452–465. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  20. Freeman, D., Scott, M., Teske, E.: A Taxonomy of Pairing-Friendly Elliptic Curves. Journal of Cryptology 23(2), 224–280 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Fouquet, M., Morain, F.: Isogeny Volcanoes and the SEA Algorithm. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 276–291. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  22. Galbraith, S.D., McKee, J., Valença, P.: Ordinary Abelian Varieties Having Small Embedding Degree. Finite Fields and Their Applications 13, 800–814 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gentry, C.: Practical Identity-Based Encryption Without Random Oracles. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 445–464. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  24. Hess, F., Smart, N., Vercauteren, F., Berlin, T.U.: The Eta Pairing Revisited. IEEE Transactions on Information Theory 52, 4595–4602 (2006)

    Article  MATH  Google Scholar 

  25. Hisil, H., Wong, K.K.-H., Carter, G., Dawson, E.: Twisted Edwards Curves Revisited. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 326–343. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  26. Ionica, S., Joux, A.: Another Approach to Pairing Computation in Edwards Coordinates. In: Chowdhury, D.R., Rijmen, V., Das, A. (eds.) INDOCRYPT 2008. LNCS, vol. 5365, pp. 400–413. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  27. 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)

    Chapter  Google Scholar 

  28. Koblitz, N., Menezes, A.: 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)

    Chapter  Google Scholar 

  29. Kohel, D.: Endomorphism Rings of Elliptic Curves over Finite Fields. PhD thesis, University of California at Berkeley (1996)

    Google Scholar 

  30. Miyaji, A., Nakabayashi, M., Takano, S.: New Explicit Conditions of Elliptic Curve traces for FR-reduction. IEICE Transactions on Fundamentals E84-A(5), 1234–1243 (2001)

    Google Scholar 

  31. Morain, F.: Edwards Curves and CM Curves (2009), http://arxiv.org/PS_cache/arxiv/pdf/0904/0904.2243v1.pdf

  32. Sahai, A., Waters, B.: Fuzzy Identity-Based Encryption. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 457–473. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  33. Sakai, R., Ohgishi, K., Kasahara, M.: Cryptosystems based on Pairing. In: SCIS 2000 (2000)

    Google Scholar 

  34. Scott, M., Barreto, P.S.L.M.: Generating more MNT Elliptic Curves. Designs, Codes and Cryptography 38, 209–217 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  35. Scott, M.: Computing the Tate Pairing. In: Menezes, A. (ed.) CT-RSA 2005. LNCS, vol. 3376, pp. 293–304. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  36. Scott, M.: On the Efficient Implementation of Pairing-Based Protocols. In: Chen, L. (ed.) IMACC 2011. LNCS, vol. 7089, pp. 296–308. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  37. Scott, M., Benger, N., Charlemagne, M., Dominguez Perez, L.J., Kachisa, E.J.: On the Final Exponentiation for Calculating Pairings on Ordinary Elliptic Curves. In: Shacham, H., Waters, B. (eds.) Pairing 2009. LNCS, vol. 5671, pp. 78–88. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  38. Tanaka, S., Nakamula, K.: Constructing Pairing-Friendly Elliptic Curves Using Factorization of Cyclotomic Polynomials. In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 136–145. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  39. Vélu, J.: Isogenies entre courbes elliptiques. Comptes Rendus De L’Academie Des Sciences Paris, Serie I-Mathematique, Serie A 273, 238–241 (1971)

    MATH  Google Scholar 

  40. Waters, B.: Efficient Identity-Based Encryption Without Random Oracles. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 114–127. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  41. Waters, B.: Ciphertext-Policy Attribute-Based Encryption: An Expressive, Efficient, and Provably Secure Realization. In: Catalano, D., Fazio, N., Gennaro, R., Nicolosi, A. (eds.) PKC 2011. LNCS, vol. 6571, pp. 53–70. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Systems, Information Technologies and Nanotechnologies, Japan

    Takanori Yasuda & Kouichi Sakurai

  2. Institute of Mathematics for Industry, Kyushu University, Japan

    Tsuyoshi Takagi

  3. Department of Informatics, Kyushu University, Japan

    Kouichi Sakurai

Authors
  1. Takanori Yasuda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tsuyoshi Takagi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kouichi Sakurai
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Research Institute for Secure Systems (RISEC), National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Umezono, 305-8568, Tsukuba, Japan

    Goichiro Hanaoka

  2. Graduate School of Natural Science and Technology, Okayama University, 3-1-1 Tsushima-naka, 700-8530, Okayama, Japan

    Toshihiro Yamauchi

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Yasuda, T., Takagi, T., Sakurai, K. (2012). Application of Scalar Multiplication of Edwards Curves to Pairing-Based Cryptography. In: Hanaoka, G., Yamauchi, T. (eds) Advances in Information and Computer Security. IWSEC 2012. Lecture Notes in Computer Science, vol 7631. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34117-5_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-34117-5_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-34116-8

  • Online ISBN: 978-3-642-34117-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation