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International Conference on Computational Science and Its Applications

ICCSA 2018: Computational Science and Its Applications – ICCSA 2018 pp 246–260Cite as

Efficiencies in Binary Elliptic Curves

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10964))

Abstract

This paper discusses the choices of elliptic curve models available to the would-be implementer, and assists the decision as to which model to use by examining the links between security and efficiency. In early public key cryptography schemes, such as ElGamal and RSA, the use of finite fields over large prime numbers was prevalent, thus preventing the need for difficult and expensive computations over extension fields. Thus, with the introduction of elliptic curve models, the same computational infrastructure using prime fields was inevitably used. As it became clear that elliptic curve models were more efficient than their public key competitors, they acquired a great deal of attention. In more recent times, and with the onset of the Internet of Things, the cryptography community is faced with the challenge of improving the efficiency of cryptography even further, resulting in many papers dealing with improvements of computational efficiencies. This search, along with improvements in both software and hardware dealing with characteristic two fields has instigated the analysis of elliptic curve constructions over binary extension fields. In particular, the ability to identify an object in the field with a bit string aids computation for binary elliptic curves. These circumstances account for our focus on binary elliptic curve fields in this paper in which we present an in-depth discussion on their efficiency and security properties along with other relevant features of various binary elliptic curve models.

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References

  1. Koblitz, N.: Elliptic curve cryptosystems. Math. Comput. 48(177), 203–209 (1987)

    Article  MathSciNet  Google Scholar 

  2. Miller, V.S.: Use of elliptic curves in cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–426. Springer, Heidelberg (1986). https://doi.org/10.1007/3-540-39799-X_31

    Chapter  Google Scholar 

  3. Lenstra Jr., H.W.: Factoring integers with elliptic curves. Ann. Math. 126(3), 649–673 (1987)

    Article  MathSciNet  Google Scholar 

  4. Fan, J., Guo, X., De Mulder, E., Schaumont, P., Preneel, B. and Verbauwhede, I.: State-of-the-art of secure ECC implementations: a survey on known side-channel attacks and countermeasures. In: 2010 IEEE International Symposium on Hardware-Oriented Security and Trust (HOST), pp. 76–87. IEEE, June 2010

    Google Scholar 

  5. Fan, J., Verbauwhede, I.: An updated survey on secure ECC implementations: attacks, countermeasures and cost. In: Naccache, D. (ed.) Cryptography and Security: From Theory to Applications. LNCS, vol. 6805, pp. 265–282. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-28368-0_18

    Chapter  MATH  Google Scholar 

  6. De Win, E., Mister, S., Preneel, B., Wiener, M.: On the performance of signature schemes based on elliptic curves. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 252–266. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0054867

    Chapter  Google Scholar 

  7. Joye, M., Yen, S.-M.: The Montgomery powering ladder. In: Kaliski, B.S., Koç, K., Paar, C. (eds.) CHES 2002. LNCS, vol. 2523, pp. 291–302. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-36400-5_22

    Chapter  Google Scholar 

  8. Marzouqi, H., Al-Qutayri, M., Salah, K.: Review of elliptic curve cryptography processor designs. Microprocess. Microsyst. 39(2), 97–112 (2015)

    Article  Google Scholar 

  9. Belgarric, P., Fouque, P.-A., Macario-Rat, G., Tibouchi, M.: Side-channel analysis of Weierstrass and Koblitz curve ECDSA on android smartphones. In: Sako, K. (ed.) CT-RSA 2016. LNCS, vol. 9610, pp. 236–252. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-29485-8_14

    Chapter  Google Scholar 

  10. Joye, M.: Highly regular right-to-left algorithms for scalar multiplication. In: Paillier, P., Verbauwhede, I. (eds.) CHES 2007. LNCS, vol. 4727, pp. 135–147. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-74735-2_10

    Chapter  Google Scholar 

  11. Montgomery, P.L.: Modular multiplication without trial division. Math. Comput. 44, 519–521 (1985)

    Article  MathSciNet  Google Scholar 

  12. Montgomery, P.L.: Speeding the Pollard and elliptic curve methods of factorization. Math. Comput. 48, 243–264 (1987)

    Article  MathSciNet  Google Scholar 

  13. Cohen, H., Frey, G., Avanzi, R., Doche, C., Lange, T., Nguyen, K., Vercauteren, F.: Handbook of Elliptic and Hyperelliptic Curve Cryptography. Chapman and Hall, CRC Press, Boca Raton (2006)

    MATH  Google Scholar 

  14. Karaklajić, D., Fan, J., Schmidt, J.M., Verbauwhede, I.: Low-cost fault detection method for ECC using Montgomery powering ladder. In: Proceedings of 2011 Design, Automation & Test in Europe, pp. 1–6. IEEE (2011)

    Google Scholar 

  15. Naccache, D., Smart, N.P., Stern, J.: Projective coordinates leak. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 257–267. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24676-3_16

    Chapter  Google Scholar 

  16. Edwards, H.: A normal form for elliptic curves. Bull. Am. Math. Soc. 44(3), 393–422 (2007)

    Article  MathSciNet  Google Scholar 

  17. Bernstein, D.J., Lange, T., Rezaeian Farashahi, R.: Binary edwards curves. In: Oswald, E., Rohatgi, P. (eds.) CHES 2008. LNCS, vol. 5154, pp. 244–265. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-85053-3_16

    Chapter  Google Scholar 

  18. Oliveira, T., López, J., Aranha, D.F., Rodríguez-Henríquez, F.: Lambda coordinates for binary elliptic curves. In: Bertoni, G., Coron, J.-S. (eds.) CHES 2013. LNCS, vol. 8086, pp. 311–330. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40349-1_18

    Chapter  Google Scholar 

  19. Kim, K.H., Lee, C.O., Negre, C.: Binary edwards curves revisited. In: Meier, W., Mukhopadhyay, D. (eds.) INDOCRYPT 2014. LNCS, vol. 8885, pp. 393–408. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-13039-2_23

    Chapter  Google Scholar 

  20. Rashidi, B.: A Survey on Hardware Implementations of Elliptic Curve Cryptosystems. arXiv preprint arXiv:1710.08336 (2017)

  21. Bernstein, D.J.: Batch binary Edwards. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 317–336. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03356-8_19

    Chapter  Google Scholar 

  22. Devigne, J., Joye, M.: Binary Huff curves. In: Kiayias, A. (ed.) CT-RSA 2011. LNCS, vol. 6558, pp. 340–355. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-19074-2_22

    Chapter  Google Scholar 

  23. Blake, I.F., Seroussi, G., Smart, N.: Elliptic curves in cryptography. In: London Mathematical Society Lecture Notes, vol. 265. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  24. [X9.62.1999] Accredited Standards Committee X9. American national standard x9.62-1999, public key cryptography for the financial services industry: The elliptic curve digital signature algorithm (ECDSA). Draft at http://grouper.ieee.org/groups/1363/Research/Other.html

  25. Oliveira, T., López, J., Aranha, D.F., Rodríguez-Henríquez, F.: Two is the fastest prime: lambda coordinates for binary elliptic curves. J. Cryptogr. Eng. 4(1), 3–17 (2014)

    Article  Google Scholar 

  26. Costello, C., Smith, B.: Montgomery curves and their arithmetic: the case of large characteristic fields. IACR Cryptology ePrint Archive, vol. 2017, p. 212 (2017)

    Google Scholar 

  27. Oliveira, T., López, J., Rodríguez-Henríquez, F.: The Montgomery ladder on binary elliptic curves. J. Cryptogr. Eng. 1–18 (2017). https://doi.org/10.1007/s13389-017-0163-8

  28. Bernstein, D.J., Lange, T.: Montgomery curves and the Montgomery ladder. IACR Cryptology ePrint Archive (2017)

    Google Scholar 

  29. Hamburg, M.: Decaf: eliminating cofactors through point compression. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9215, pp. 705–723. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-47989-6_34

    Chapter  Google Scholar 

  30. Farashahi, R.R., Joye, M.: Efficient arithmetic on Hessian curves. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 243–260. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13013-7_15

    Chapter  Google Scholar 

  31. Solinas, J.A.: Efficient arithmetic on Koblitz curves. In: Koblitz, N. (ed.) Towards a Quarter-Century of Public Key Cryptography, pp. 125–179. Springer, Boston (2000). https://doi.org/10.1007/978-1-4757-6856-5_6

    Chapter  Google Scholar 

  32. Aranha, D.F., Faz-Hernández, A., López, J., Rodríguez-Henríquez, F.: Faster implementation of scalar multiplication on Koblitz curves. In: Hevia, A., Neven, G. (eds.) LATINCRYPT 2012. LNCS, vol. 7533, pp. 177–193. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-33481-8_10

    Chapter  Google Scholar 

  33. Bernstein, D., Lange, T.: Explicit-Formulas Database (2014). http://hyperelliptic.org/EFD/. Accessed 2 Apr 2017

  34. Gallant, R.P., Lambert, R.J., Vanstone, S.A.: Faster point multiplication on elliptic curves with efficient endomorphisms. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 190–200. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44647-8_11

    Chapter  Google Scholar 

  35. Galbraith, S.D., Lin, X., Scott, M.: Endomorphisms for faster elliptic curve cryptography on a large class of curves. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 518–535. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-01001-9_30

    Chapter  Google Scholar 

  36. Hankerson, D., Karabina, K., Menezes, A.: Analyzing the Galbraith-Lin-Scott point multiplication method for elliptic curves over binary fields. IEEE Trans. Comput. 58(10), 1411–1420 (2009)

    Article  MathSciNet  Google Scholar 

  37. Gueron, S.: AES-GCM for efficient authenticated encryption–ending the reign of HMAC-SHA-1. Real-World Cryptography (2013)

    Google Scholar 

  38. Alcaide, A., Palomar, E., Montero-Castillo, J., Ribagorda, A.: Anonymous authentication for privacy-preserving IoT target-driven applications. Comput. Secur. 37, 111–123 (2013)

    Article  Google Scholar 

  39. Markmann, T., Schmidt, T.C., Wählisch, M.: Federated end-to-end authentication for the constrained internet of things using IBC and ECC. ACM SIGCOMM Comput. Commun. Rev. 45(4), 603–604 (2015)

    Article  Google Scholar 

  40. Chatzigiannakis, I., Vitaletti, A., Pyrgelis, A.: A privacy-preserving smart parking system using an IoT elliptic curve based security platform. Comput. Commun. 89, 165–177 (2016)

    Article  Google Scholar 

  41. Wenger, E., Hutter, M.: Exploring the design space of prime field vs. binary field ECC-hardware implementations. In: Laud, P. (ed.) NordSec 2011. LNCS, vol. 7161, pp. 256–271. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29615-4_18

    Chapter  Google Scholar 

  42. Azarderakhsh, R., Jarvinen, K.U., Mozaffari-Kermani, M.: Efficient algorithm and architecture for elliptic curve cryptography for extremely constrained secure applications. IEEE Trans. Circ. Syst. I Regul. Pap. 61(4), 1144–1155 (2014)

    Article  Google Scholar 

  43. Halak, B., Waizi, S.S., Islam, A.: A Survey of Hardware Implementations of Elliptic Curve Cryptographic Systems (2016). https://eprint.iacr.org/2016/712.pdf

  44. Ozturk, E., Gopal, V.: Enabling High-performance Galois-counter mode on Intel architecture processors. Intel white paper (2012)

    Google Scholar 

  45. Galbraith, S.D., Gaudry, P.: Recent progress on the elliptic curve discrete logarithm problem. Des. Codes Crypt. 78(1), 51–72 (2016)

    Article  MathSciNet  Google Scholar 

  46. Feix, B., Roussellet, M., Venelli, A.: Side-channel analysis on blinded regular scalar multiplications. In: Meier, W., Mukhopadhyay, D. (eds.) INDOCRYPT 2014. LNCS, vol. 8885, pp. 3–20. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-13039-2_1

    Chapter  Google Scholar 

  47. Chen, C.: FPGA implementation for elliptic curve cryptography over binary extension field. M.A.Sc., University of Windsor, 10 December 2017, Electronic Theses and Dissertations (2017)

    Google Scholar 

  48. Lalonde, D.R.: Private and public-key side-channel threats against hardware accelerated cryptosystems. M.A.Sc., University of Windsor, 13 December 2017, Electronic Theses and Dissertations (2017)

    Google Scholar 

  49. Coron, J.-S.: Resistance against differential power analysis for elliptic curve cryptosystems. In: Koç, Ç.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 292–302. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48059-5_25

    Chapter  Google Scholar 

  50. Maplesoft. User Manual (2015). http://www.maplesoft.com/documentation_center/

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Authors and Affiliations

  1. Deakin University, Geelong, VIC, 3216, Australia

    Scott T. E. Hirschfeld, Lynn M. Batten & Mohammed K. I. Amain

Authors
  1. Scott T. E. Hirschfeld
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  2. Lynn M. Batten
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  3. Mohammed K. I. Amain
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Correspondence to Scott T. E. Hirschfeld .

Editor information

Editors and Affiliations

  1. University of Perugia, Perugia, Italy

    Osvaldo Gervasi

  2. University of Basilicata, Potenza, Italy

    Beniamino Murgante

  3. Covenant University, Ota, Nigeria

    Sanjay Misra

  4. Saint Petersburg State University, Saint Petersburg, Russia

    Elena Stankova

  5. Polytechnic University of Bari, Bari, Italy

    Carmelo M. Torre

  6. University of Minho, Braga, Portugal

    Ana Maria A.C. Rocha

  7. Monash University, Clayton, Victoria, Australia

    David Taniar

  8. Kyushu Sangyo University, Fukuoka shi, Fukuoka, Japan

    Bernady O. Apduhan

  9. Politecnico di Bari, Bari, Italy

    Eufemia Tarantino

  10. Myongji University, Yongin, Korea (Republic of)

    Yeonseung Ryu

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Hirschfeld, S.T.E., Batten, L.M., Amain, M.K.I. (2018). Efficiencies in Binary Elliptic Curves. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2018. ICCSA 2018. Lecture Notes in Computer Science(), vol 10964. Springer, Cham. https://doi.org/10.1007/978-3-319-95174-4_21

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  • DOI: https://doi.org/10.1007/978-3-319-95174-4_21

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