Skip to main content
SpringerLink
Log in

A New Family of Pairing-Friendly Elliptic Curves

  • Conference paper
  • First Online:
Arithmetic of Finite Fields (WAIFI 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11321))

Included in the following conference series:

Abstract

There have been recent advances in solving the finite extension field discrete logarithm problem as it arises in the context of pairing-friendly elliptic curves. This has lead to the abandonment of approaches based on supersingular curves of small characteristic, and to the reconsideration of the field sizes required for implementation based on non-supersingular curves of large characteristic. This has resulted in a revision of recommendations for suitable curves, particularly at a higher level of security. Indeed for a security level of 256 bits, the BLS48 curves have been suggested, and demonstrated to be superior to other candidates. These curves have an embedding degree of 48. The well known taxonomy of Freeman, Scott and Teske only considered curves with embedding degrees up to 50. Given some uncertainty around the constants that apply to the best discrete logarithm algorithm, it would seem to be prudent to push a little beyond 50. In this note we announce the discovery of a new family of pairing friendly elliptic curves which includes a new construction for a curve with an embedding degree of 54.

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

Access this chapter

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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

Institutional subscriptions

Notes

  1. 1.

    http://www.cerias.purdue.edu/homes/ssw/cun/index.html.

References

  1. Barbulescu, R., Duquesne, S.: Updating key size estimations for pairings. J. Cryptol. (2018). https://doi.org/10.1007/s00145-018-9280-5. http://eprint.iacr.org/2017/334

  2. Barreto, P.S.L.M., Lynn, B., Scott, M.: Constructing elliptic curves with prescribed embedding degrees. In: Cimato, S., Persiano, G., Galdi, C. (eds.) SCN 2002. LNCS, vol. 2576, pp. 257–267. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-36413-7_19

    Chapter  Google Scholar 

  3. 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). https://doi.org/10.1007/11693383_22

    Chapter  Google Scholar 

  4. 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). https://doi.org/10.1007/978-3-642-13797-6_13

    Chapter  MATH  Google Scholar 

  5. Brent, R.P.: On computing factors of cyclotomic polynomials. Math. Comp. 61(203), 131–149 (1993). https://doi.org/10.2307/2152941

    Article  MathSciNet  MATH  Google Scholar 

  6. Brezing, F., Weng, A.: Elliptic curves suitable for pairing based cryptography. Des. Codes Cryptogr. 37(1), 133–141 (2005). https://eprint.iacr.org/2003/143

    Article  MathSciNet  Google Scholar 

  7. Brillhart, J., Lehmer, D.H., Selfridge, J.L., Tuckerman, B., Wagstaff Jr., S.S.: Factorizations of \(b^n \pm 1\), \(b=2,3,5,6,7,10,11,12\) up to High Powers. Contemporary Mathematics, 2nd edn, vol. 22. American Mathematical Society, Providence (1988). https://homes.cerias.purdue.edu/ssw/cun/

  8. Estibals, N.: Compact hardware for computing the tate pairing over 128-bit-security supersingular curves. In: Joye, M., Miyaji, A., Otsuka, A. (eds.) Pairing 2010. LNCS, vol. 6487, pp. 397–416. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-17455-1_25

    Chapter  MATH  Google Scholar 

  9. Freeman, D., Scott, M., Teske, E.: A taxonomy of pairing-friendly elliptic curves. J. Cryptol. 23(2), 224–280 (2010). http://eprint.iacr.org/2006/372

    Article  MathSciNet  Google Scholar 

  10. Galbraith, S.D., McKee, J.F., Valença, P.C.: Ordinary Abelian varieties having small embedding degree. Finite Fields Appl. 13(4), 800–814 (2007). https://eprint.iacr.org/2004/365

    Article  MathSciNet  Google Scholar 

  11. Granville, A., Pleasants, P.: Aurifeuillian factorization. Math. Comp. 75(253), 497–508 (2006). https://doi.org/10.1090/S0025-5718-05-01766-7

    Article  MathSciNet  MATH  Google Scholar 

  12. Joux, A., Pierrot, C.: The special number field sieve in \(\mathbb{F}_{p^{n}}\) - application to pairing-friendly constructions. In: Cao, Z., Zhang, F. (eds.) Pairing 2013. LNCS, vol. 8365, pp. 45–61. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-04873-4_3

    Chapter  MATH  Google Scholar 

  13. 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). https://doi.org/10.1007/978-3-540-85538-5_9

    Chapter  MATH  Google Scholar 

  14. Kim, T., Barbulescu, R.: Extended tower number field sieve: a new complexity for the medium prime case. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9814, pp. 543–571. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53018-4_20

    Chapter  Google Scholar 

  15. Kiyomura, Y., Inoue, A., Kawahara, Y., Yasuda, M., Takagi, T., Kobayashi, T.: Secure and efficient pairing at 256-bit security level. In: Gollmann, D., Miyaji, A., Kikuchi, H. (eds.) ACNS 2017. LNCS, vol. 10355, pp. 59–79. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-61204-1_4

    Chapter  Google Scholar 

  16. Menezes, A., Sarkar, P., Singh, S.: Challenges with assessing the impact of NFS advances on the security of pairing-based cryptography. In: Phan, R.C.-W., Yung, M. (eds.) Mycrypt 2016. LNCS, vol. 10311, pp. 83–108. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-61273-7_5

    Chapter  Google Scholar 

  17. El Mrabet, N., Joye, M. (eds.): Guide to Pairing-Based Cryptography. Chapman and Hall/CRC, Boca Raton (2016). https://www.crcpress.com/Guide-to-Pairing-Based-Cryptography/El-Mrabet-Joye/p/book/9781498729505

  18. Schinzel, A.: On primitive prime factors of \(a^n-b^n\). Proc. Cambridge Philos. Soc. 58(4), 555–562 (1962). https://doi.org/10.1017/S0305004100040561

    Article  MathSciNet  MATH  Google Scholar 

  19. Schirokauer, O.: The number field sieve for integers of low weight. Math. Comput. 79(269), 583–602 (2010). https://doi.org/10.1090/S0025-5718-09-02198-X. http://eprint.iacr.org/2006/107

    Article  MathSciNet  Google Scholar 

  20. 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). https://doi.org/10.1007/978-3-642-25516-8_18

    Chapter  Google Scholar 

  21. 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). https://doi.org/10.1007/978-3-642-03298-1_6

    Chapter  MATH  Google Scholar 

  22. Stevenhagen, P.: On Aurifeuillian factorizations. Nederl. Akad. Wetensch. Indag. Math. 49(4), 451–468 (1987). https://doi.org/10.1016/1385-7258(87)90009-6

    Article  MathSciNet  MATH  Google Scholar 

  23. Vercauteren, F.: Optimal pairings. IEEE Trans. Inf. Theory 56, 455–461 (2009). https://eprint.iacr.org/2008/096

    Article  MathSciNet  Google Scholar 

  24. Wagstaff Jr., S.S.: The search for Aurifeuillian-like factorizations. J. Integers 12A(6), 1449–1461 (2012). https://homes.cerias.purdue.edu/~ssw/cun/mine.pdf

Download references

Author information

Authors and Affiliations

  1. MIRACL.com, Trim, Ireland

    Michael Scott

  2. Université de Lorraine, CNRS, Inria, LORIA, Nancy, France

    Aurore Guillevic

Authors
  1. Michael Scott
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Aurore Guillevic
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Scott .

Editor information

Editors and Affiliations

  1. Department of Informatics, University of Bergen, Bergen, Norway

    Lilya Budaghyan

  2. Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, Mexico, Mexico

    Francisco Rodríguez-Henríquez

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Scott, M., Guillevic, A. (2018). A New Family of Pairing-Friendly Elliptic Curves. In: Budaghyan, L., Rodríguez-Henríquez, F. (eds) Arithmetic of Finite Fields. WAIFI 2018. Lecture Notes in Computer Science(), vol 11321. Springer, Cham. https://doi.org/10.1007/978-3-030-05153-2_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-05153-2_2

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-05152-5

  • Online ISBN: 978-3-030-05153-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation