Abstract
We provide several results on some families of twisted Jacobi quartics. We give new addition formulæ for two models of twisted Jacobi quartic elliptic curves, which represent respectively 1/6 and 2/3 of all elliptic curves, with respective costs 7M + 3S + D a and 8M + 3 S + D a . These formulæ allow addition and doubling of points, except for points differing by a point of order two.
Furthermore, we give an intrinsic characterization of elliptic curves represented by the classical Jacobi quartic, by the action of the Frobenius endomorphism on the 4-torsion subgroup. This allows us to compute the exact proportion of elliptic curves representable by various models (the three families of Jacobi quartics, plus Edwards and Huff curves) from statistics on this Frobenius action.
This work was supported by the French Agence Nationale de la Recherche through the ECLIPSES project under Contract ANR-09-VERS-018.
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References
Ahmadi, O., Granger, R.: On isogeny classes of edwards curves over finite fields. Arxiv preprint arXiv:1103.3381 (2011)
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), doi:10.1007/978-3-540-68164-9_26
Billet, O., Joye, M.: The Jacobi Model of an Elliptic Curve and Side-Channel Analysis. In: Fossorier, M., Høholdt, T., Poli, A. (eds.) AAECC 2003. LNCS, vol. 2643, pp. 34–42. Springer, Heidelberg (2003), doi:10.1007/3-540-44828-4_5
Bernstein, D.J., Lange, T.: Inverted Edwards Coordinates. In: Boztaş, S., Lu, H.-F. (eds.) AAECC 2007. LNCS, vol. 4851, pp. 20–27. Springer, Heidelberg (2007)
Castryck, W., Hubrechts, H.: The distribution of the number of points modulo an integer on elliptic curves over finite fields (Preprint, 2011)
Doche, C., Icart, T., Kohel, D.R.: Efficient Scalar Multiplication by Isogeny Decompositions. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T. (eds.) PKC 2006. LNCS, vol. 3958, pp. 191–206. Springer, Heidelberg (2006)
Edwards, H.M.: A normal form for elliptic curves. Bulletin-American Mathematical Society 44(3), 393–422 (2007)
Feng, R., Wu, H.: On the isomorphism classes of legendre elliptic curves over finite fields. Arxiv preprint arXiv:1001.2871 (2010)
Gekeler, E.-U.: The distribution of group structures on elliptic curves over finite prime fields. Documenta Mathematica 11, 119–142 (2006)
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)
Hisil, H., Wong, K.K.-H., Carter, G., Dawson, E.: Faster group operations on elliptic curves. In: Proceedings of the Seventh Australasian Conference on Information Security, AISC 2009, vol. 98, pp. 7–20. Australian Computer Society, Inc., Darlinghurst (2009)
Joye, M., Tibouchi, M., Vergnaud, D.: Huff’s Model for Elliptic Curves. In: Hanrot, G., Morain, F., Thomé, E. (eds.) ANTS-IX. LNCS, vol. 6197, pp. 234–250. Springer, Heidelberg (2010)
Liardet, P.-Y., Smart, N.P.: Preventing SPA/DPA in ECC Systems using the Jacobi Form. In: Koç, Ç.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 391–401. Springer, Heidelberg (2001), doi:10.1007/3-540-44709-1_32
Morain, F.: Edwards curves and cm curves. Arxiv preprint arXiv:0904.2243 (2009)
National Institute of Standards and Technology. FIPS PUB 186-2: Digital Signature Standard (DSS) (January 2000)
Ono, T.: Variations on a theme of Euler: quadratic forms, elliptic curves, and Hopf maps. Plenum. Pub. Corp. (1994)
Farashahi, R.R., Shparlinski, I.: On the number of distinct elliptic curves in some families. Designs, Codes and Cryptography 54, 83–99 (2010), doi:10.1007/s10623-009-9310-2
Serre, J.-P.: Zeta and L functions. In: Proc. Conf. on Arithmetical Algebraic Geometry, Purdue Univ., pp. 82–92. Harper & Row, New York (1965)
Silverman, J.H.: The arithmetic of elliptic curves. Springer, Heidelberg (1986)
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Plût, J. (2012). On Various Families of Twisted Jacobi Quartics. In: Miri, A., Vaudenay, S. (eds) Selected Areas in Cryptography. SAC 2011. Lecture Notes in Computer Science, vol 7118. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28496-0_22
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