Twisted Hessian Curves

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9230)

Abstract

This paper presents new speed records for arithmetic on a large family of elliptic curves with cofactor 3: specifically, \(8.77\mathbf{M}\) per bit for 256-bit variable-base single-scalar multiplication when curve parameters are chosen properly. This is faster than the best results known for cofactor 1, showing for the first time that points of order 3 are useful for performance and narrowing the gap to the speeds of curves with cofactor 4.

Keywords

Efficiency Elliptic-curve arithmetic Double-base chains Fast arithmetic Hessian curves Complete addition laws 

References

  1. 1.
    Aréne, C., Lange, T., Naehrig, M., Ritzenthaler, C.: Faster computation of the Tate pairing. J. Number Theor. 131, 842–857 (2011)CrossRefMATHGoogle Scholar
  2. 2.
    Aronhold, S.H.: Zur Theorie der homogenen Functionen dritten Grades von drei Variabeln. Crelles J. für die reine und angewandte Mathematik 1850(39), 140–159 (1850)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Benaloh, J. (ed.): CT-RSA 2014. LNCS, vol. 8366. Springer, Heidelberg (2014)Google Scholar
  4. 4.
    Bernstein, D.J.: Complete addition laws for all elliptic curves over finite fields (talk slides) (2009). http://cr.yp.to/talks/2009.07.17/slides.pdf
  5. 5.
    Bernstein, D.J.: Curve25519: new Diffie-Hellman speed records. In: PKC 2006 [52], pp. 207–228 (2006)Google Scholar
  6. 6.
    Bernstein, D.J., Birkner, P., Lange, T., Peters, C.: Optimizing double-base ellipticcurve single-scalar multiplication. In: Indocrypt 2007 [51], pp. 167–182 (2007)Google Scholar
  7. 7.
    Bernstein, D.J., Duif, N., Lange, T., Schwabe, P., Yang, B.-Y.: High-speed high-security signatures. J. Cryptographic Eng. 2, 77–89 (2012)CrossRefGoogle Scholar
  8. 8.
    Bernstein, D.J., Lange, T.: Explicit-formulas database (2007). https://hyperelliptic.org/EFD
  9. 9.
    Bernstein, D.J., Lange, T.: Faster addition and doubling on elliptic curves. In: Asiacrypt 2007 [40], pp. 29–50 (2007)Google Scholar
  10. 10.
    Bernstein, D.J., Lange, T.: Analysis and optimization of elliptic-curve single-scalar multiplication, In: Fq8 [44], pp. 1–19 (2008)Google Scholar
  11. 11.
    Bernstein, D.J., Lange, T.: A complete set of addition laws for incomplete Edwards curves. J. Number Theor. 131, 858–872 (2011)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Bertoni, G., Coron, J.-S. (eds.): CHES 2013. LNCS, vol. 8086, pp. 142–158. Springer, Heidelberg (2013) CrossRefMATHGoogle Scholar
  13. 13.
    Billet, O., Joye, M.: The Jacobi model of an elliptic curve and side-channel analysis. In: AAECC 2003 [28], pp. 34–42 (2003)Google Scholar
  14. 14.
    Bosma, W., Lenstra Jr., H.W.: Complete systems of two addition laws for elliptic curves. J. Number Theor. 53, 229–240 (1995)Google Scholar
  15. 15.
    Brankovic, L., Susilo, W. (eds.): Australasian information security conference (AISC 2009), Wellington, New Zealand, January 2009. In: Conferences in Research and Practice in Information Technology (CRPIT), 1998. Australian Computer Society Inc. (2009)Google Scholar
  16. 16.
    Cayley, A.: On the 34 concomitants of the ternary cubic. Am. J. Math. 4, 1–15 (1881)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Chudnovsky, D.V., Chudnovsky, G.V.: Sequences of numbers generated by addition in formal groups and new primality and factorization tests. Adv. Appl. Math. 7, 385–434 (1986)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Cohen, H., Frey, G. (eds.): Handbook of elliptic and hyperelliptic curve cryptography. CRC Press (2005)Google Scholar
  19. 19.
    Cohen, H., Miyaji, A., Ono, T.: Efficient elliptic curve exponentiation using mixed coordinates. In: Asiacrypt 1998 [48], pp. 51–65 (1998)Google Scholar
  20. 20.
    Costello, C., Hisil, H., Smith, B.: Faster compact Diffie–Hellman: endomorphismson the x-line. In: Eurocrypt 2014 [45], pp. 183–200 (2014)Google Scholar
  21. 21.
    Doche, C., Habsieger, L.: A tree-based approach for computing double-base chains. In: ACISP 2008 [43], pp. 433–446 (2008)Google Scholar
  22. 22.
    Doche, C., Icart, T., Kohel, D.R.: Efficient scalar multiplication by isogeny decompositions. In: PKC 2006 [52], pp. 191–206 (2006)Google Scholar
  23. 23.
    Doche, C., Lange, T.: Arithmetic of elliptic curves. In: HEHCC [18], pp. 267–302 (2005)Google Scholar
  24. 24.
    Edwards, H.M.: A normal form for elliptic curves. Bull. Am. Math. Soc. 44, 393–422 (2007)CrossRefMATHGoogle Scholar
  25. 25.
    Farashahi, R.R., Joye, M.: Efficient arithmetic on Hessian curves. In: PKC 2010 [46], pp. 243–260 (2010)Google Scholar
  26. 26.
    Farashahi, R.R., Wu, H., Zhao, C.-A.: Efficient Arithmetic on Elliptic Curves over Fields of Characteristic Three. In: SAC 2012 [35], pp. 135–148 (2013)Google Scholar
  27. 27.
    Faz-Hernández, A., Longa, P., Sánchez, A.H.: Efficient and Secure Algorithms for GLV-Based Scalar Multiplication and Their Implementation on GLV-GLS Curves. In: CT-RSA 2014 [3], pp. 1–27 (2014)Google Scholar
  28. 28.
    Fossorier, M.P.C., Høholdt, T., Poli, A. (eds.): AAECC 2003. LNCS, vol. 2643. Springer, Heidelberg (2003) MATHGoogle Scholar
  29. 29.
    Hesse, O.: Über die Elimination der Variabeln aus drei algebraischen Gleichungen vom zweiten Grade mit zwei Variabeln. J. für die Reine und Angewandte Mathematik 28, 68–96 (1844)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Hisil, H., Carter, G., Dawson, E.: New formulae for efficient elliptic curve arithmetic. In: Indocrypt 2007 [51], pp. 138–151 (2007)Google Scholar
  31. 31.
    Hisil, H., Wong, K.K-H., Carter, G., Dawson, E.: Faster group operations on elliptic curve. In: AISC 2009 [15], pp. 7–19 (2009)Google Scholar
  32. 32.
    Hisil, H.: Elliptic curves, group law, and efficient computation, Ph.D. thesis, Queensland University of Technology (2010)Google Scholar
  33. 33.
    Husemöller, D.: Elliptic Curves. Graduate Texts in Mathematics, vol. 111, 2nd edn. Springer, New York (2003)Google Scholar
  34. 34.
    Joye, M., Quisquater, J.-J.: Hessian Elliptic Curves and Side-Channel Attacks. In: CHES 2001 [37], pp. 402–410 (2001)Google Scholar
  35. 35.
    Knudsen, L.R., Wu, H. (eds.): SAC 2012. LNCS, vol. 7707. Springer, Heidelberg (2013) Google Scholar
  36. 36.
    Koblitz, N.: Algebraic Aspects of Cryptography. Algorithms and Computation in Mathematics, vol. 3. Springer, Heidelberg (1998) MATHGoogle Scholar
  37. 37.
    Koç, Ç.K., Naccache, D., Paar, C. (eds.): CHES 2001. LNCS, vol. 2162. Springer, Heidelberg (2001) MATHGoogle Scholar
  38. 38.
    Kohel, D.: Addition law structure of elliptic curves. J. Number Theor. 131, 894–919 (2011)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Kohel, D.: The geometry of efficient arithmetic on elliptic curves. In: Arithmetic, Geometry, Coding Theory and Cryptography, vol. 637. pp. 95–109 (2015)Google Scholar
  40. 40.
    Kurosawa, K. (ed.): ASIACRYPT 2007. LNCS, vol. 4833. Springer, Heidelberg (2007) MATHGoogle Scholar
  41. 41.
    Liardet, P.-Y., Smart, N.P.: Preventing SPA/DPA in ECC Systems Using the Jacobi Form. In: CHES 2001 [37], pp. 391–401 (2001)Google Scholar
  42. 42.
    Montgomery, P.L.: Speeding the Pollard and elliptic curve methods of factorization. Math. Comput. 48, 243–264 (1987)CrossRefMATHGoogle Scholar
  43. 43.
    Mu, Y., Susilo, W., Seberry, J. (eds.): ACISP 2008. LNCS, vol. 5107. Springer, Heidelberg (2008) MATHGoogle Scholar
  44. 44.
    Mullen, G.L., Panario, D., Shparlinski, I.E. (eds.): Finite Fields and applications. In: papers from the 8th international conference held in Melbourne, July 9–13, 2007, Contemporary Mathematics, 461, American Mathematical Society (2008)Google Scholar
  45. 45.
    Nguyen, P.Q., Oswald, E. (eds.): EUROCRYPT 2014. LNCS, vol. 8441. Springer, Heidelberg (2014) MATHGoogle Scholar
  46. 46.
    Nguyen, P.Q., Pointcheval, D. (eds.): PKC 2010. LNCS, vol. 6056. Springer, Heidelberg (2010) MATHGoogle Scholar
  47. 47.
    National Institute of Standards and Technology: Recommended elliptic curves for federal government use (1999). http://csrc.nist.gov/groups/ST/toolkit/documents/dss/NISTReCur.pdf
  48. 48.
    Ohta, K., Pei, D. (eds.): ASIACRYPT 1998. LNCS, vol. 1514. Springer, Heidelberg (1998) MATHGoogle Scholar
  49. 49.
    Oliveira, T., López, J., Aranha, D.F., Rodríguez-Henríquez, F.: Lambda Coordinates for Binary Elliptic Curves. In: CHES 2013 [12], pp. 311–330 (2013)Google Scholar
  50. 50.
    Smart, N.P.: The Hessian Form of an Elliptic Curve. In: CHES 2001 [37], pp. 118–125(2001)Google Scholar
  51. 51.
    Srinathan, K., Rangan, C.P., Yung, M. (eds.): INDOCRYPT 2007. LNCS, vol. 4859. Springer, Heidelberg (2007) MATHGoogle Scholar
  52. 52.
    Yung, M., Dodis, Y., Kiayias, A., Malkin, T. (eds.): PKC 2006. LNCS, vol. 3958. Springer, Heidelberg (2006) Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceTechnische Universiteit EindhovenEindhovenThe Netherlands
  2. 2.Department of Computer ScienceUniversity of Illinois at ChicagoChicagoUSA
  3. 3.Institut de Mathématiques de MarseilleAix-Marseille UniversitéMarseille Cedex 09France

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