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Speeding up Huff form of elliptic curves

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Abstract

This paper presents faster inversion-free point addition formulas for the curve \(y (1+ax^2) = cx (1+dy^2)\). The proposed formulas improve the point doubling operation count record (I, M, S, D, a are arithmetic operations over a field. I: inversion, M: multiplication, S: squaring, D: multiplication by a curve constant, a: addition/subtraction) from \(6\mathbf{{M}}+ 5\mathbf{{S}}\) to \(8\mathbf{{M}}\) and mixed addition operation count record from \(10\mathbf{{M}}\) to \(8\mathbf{{M}}\). Both sets of formulas are shown to be 4-way parallel, leading to an effective cost of \(2\mathbf{{M}}\) per either of the group operations.

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Notes

  1. Here, “small” refers to any distinguished element for which multiplications with other field elements are significantly faster than the usual multiplication of two arbitrary elements in \(\mathbb {K}\).

  2. The singularities of E is ommitted here for simplicity. One can work with desingularization of E in \(\mathbb {P}^1\times {}\mathbb {P}^1\) to get a technically better statement.

References

  1. Bernstein D.J., Birkner P., Joye M., Lange T., Peters C.: Twisted Edwards curves. In: AFRICACRYPT 2008 Proceedings, LNCS, vol. 5023, pp. 389–405. Springer (2008).

  2. Bernstein D.J., Chuengsatiansup C., Kohel D., Lange T.: Twisted Hessian curves. In: Progress in Cryptology LATINCRYPT 2015 Proceedings, vol. 9230, pp. 269–294. Springer (2015).

  3. Bernstein D.J., Lange T.: Faster addition and doubling on elliptic curves. In: ASIACRYPT 2007, LNCS, vol. 4833, pp. 29–50. Springer (2007).

  4. Bernstein D.J., Lange T.: A complete set of addition laws for incomplete Edwards curves. J. Number Theory 131(5), 858–872 (2011).

    Article  MathSciNet  Google Scholar 

  5. Billet O., Joye M.: The Jacobi model of an elliptic curve and side-channel analysis. In: 2003 Proceedings Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: 15th International Symposium, AAECC-15, Toulouse, France, 12–16 May, vol. 2643, pp. 34–42. Springer, Berlin (2003).

    Chapter  Google Scholar 

  6. Ciss A.A., Sow D.: On a new generalization of Huff curves. Cryptology ePrint Archive, Report 2011/580, (2011). http://eprint.iacr.org/2011/580.

  7. Hankerson D., Menezes A.J., Vanstone S.A.: Guide to Elliptic Curve Cryptography. Springer, New York (2003).

    MATH  Google Scholar 

  8. Hisil H.: Elliptic curves, group law, and efficient computation. PhD thesis, Queensland University of Technology (2010).

  9. Hisil H., Wong K.K.-H., Carter G., Dawson E.: Twisted Edwards curves revisited. In: ASIACRYPT 2008 Proceedings, LNCS, vol. 5350, pp. 326–343. Springer (2008).

  10. Hisil H., Wong K.K.-H., Carter G., Dawson E.: Jacobi quartic curves revisited. In: ACISP 2009 proceedings, LNCS, vol. 5594, pp. 452–468. Springer (2009).

  11. Huff G.B.: Diophantine problems in geometry and elliptic ternary forms. Duke Math. J. 15(2), 443–453 (1948).

    Article  MathSciNet  Google Scholar 

  12. Joye M., Tibouchi M., Vergnaud D.: Huff’s model for elliptic curves. In: Algorithmic Number Theory: 9th International Symposium, ANTS-IX, Nancy, France, July 19–23, 2010 Proceedings, vol. 6197, pp. 234–250. Springer, Berlin (2010).

    Google Scholar 

  13. Moody D., Shumow D.: Analogues of vélu’s formulas for isogenies on alternate models of elliptic curves. Math. Comput. 85(300), 1929–1951 (2016).

    Article  Google Scholar 

  14. Silverman J.H.: The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, vol. 106. Springer, 1st ed. 1986. Corr. 3rd printing (1994).

  15. Wu H., Feng R.: Elliptic curves in Huff’s model. Cryptology ePrint Archive, Report 2010/383 (2010). http://eprint.iacr.org/2010/390.

  16. Wu H., Feng R.: Elliptic curves in Huff’s model. Wuhan Univ. J. Nat. Sci. 17(6), 473–480 (2012).

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors thank the anonymous reviewers for their suggestions and comments.

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Authors and Affiliations

  1. Gediz Electric Inc., Gediz, Turkey

    Neriman Gamze Orhon

  2. Yasar University, Izmir, Turkey

    Huseyin Hisil

Authors
  1. Neriman Gamze Orhon
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  2. Huseyin Hisil
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Corresponding author

Correspondence to Huseyin Hisil.

Additional information

Communicated by A. Enge.

This project is funded by Yasar University Scientific Research Project SRP-024.

Appendices

Appendix A: Maple verification script for Theorem (2)

figure e

Appendix B: Maple verification script for Theorem (3)

figure f

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Orhon, N.G., Hisil, H. Speeding up Huff form of elliptic curves. Des. Codes Cryptogr. 86, 2807–2823 (2018). https://doi.org/10.1007/s10623-018-0475-4

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