Abstract
In this paper, we present a closed formula for the Tate pairing computation for supersingular elliptic curves defined over the binary field \(\mathbb F_{2^m}\) of odd dimension. There are exactly three isomorphism classes of supersingular elliptic curves over \(\mathbb F_{2^m}\) for odd m and our result is applicable to all these curves.
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Menezes, A.J., Okamoto, T., Vanstone, S.A.: Reducing elliptic curve logarithms to logarithms in a finite field. IEEE Trans. Information Theory 39, 1639–1646 (1993)
Frey, G., Rück, H.: A remark concerning m-divisibility and the discrete logarithm in the divisor class groups of curves. Math. Comp. 62, 865–874 (1994)
Boneh, D., Franklin, M.: Identity based encryption from the Weil pairing. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 213–229. Springer, Heidelberg (2001)
Boneh, D., Lynn, B., Shacham, H.: Short signatures from the Weil pairing. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 514–532. Springer, Heidelberg (2001)
Joux, A.: A one round protocol for tripartite Diffie-Hellman. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 385–394. Springer, Heidelberg (2000)
Sakai, R., Ohgishi, K., Kasahara, M.: Cryptosystems based on pairing. In: SICS 2000, Symposium on Cryptography and Information Security, pp. 26–28 (2000)
Smart, N.P.: An identity based authentication key agreement protocol based on pairing. Electronics Letters 38, 630–632 (2002)
Granger, R., Page, D., Stam, M.: Hardware and software normal basis arithmetic for pairing based cryptography in characteristic three (preprint) (2004), available at http://eprint.iacr.org/2004/157.pdf
Granger, R., Page, D., Stam, M.: On small characteristic algebraic tori in pairing based cryptography, preprint (2004), available at http://eprint.iacr.org/2004/132.pdf
Duursma, I., Lee, H.: Tate pairing implementation for hyperelliptic curves y2 = xp − x + d. In: Laih, C.-S. (ed.) ASIACRYPT 2003. LNCS, vol. 2894, pp. 111–123. Springer, Heidelberg (2003)
Eisenträger, K., Lauter, K., Montgomery, P.L.: ImprovedWeil and Tate pairing for elliptic and hyperelliptic curves, preprint (2004)
Miyaji, A., Nakabayashi, M., Takano, S.: New explicit conditions of elliptic curve trace for FR-reduction. IEICE Trans. Fundamentals E84 A, 1–10 (2001)
Silverman, J.H.: The Arithmetic of Elliptic Curves. Springer, Heidelberg (1985)
Barreto, P., Kim, H., Lynn, B., Scott, M.: Efficient algorithms for pairing based cryptosystems. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 354–368. Springer, Heidelberg (2002)
Hess, F.: A Note on the Tate pairing of curves over finite fields. Arch. Math. 82, 28–32 (2004)
Hess, F.: Efficient identity based signature schemes based on pairings. In: Nyberg, K., Heys, H.M. (eds.) SAC 2002. LNCS, vol. 2595, pp. 310–324. Springer, Heidelberg (2003)
Menezes, A.J.: Elliptic Curve Public Key Cryptosystems. Kluwer Academic Publishers, Dordrecht (1993)
Harrison, K.: Personal Communications (2004)
Cha, J.C., Cheon, J.H.: An identity-based signature from gap Diffie-Hellman groups. In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 18–30. Springer, Heidelberg (2002)
Rubin, K., Silverberg, A.: Torus based cryptography. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 349–365. Springer, Heidelberg (2003)
Galbraith, S., Harrison, K., Soldera, D.: Implementing the Tate pairing. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 324–337. Springer, Heidelberg (2002)
Miller, V.: Short programs for functions on curves (1986) (unpublished manuscript)
Hankerson, D., Hernandez, J.L., Menezes, A.J.: Software implementation of elliptic curve cryptography over binary fields. In: Paar, C., Koç, Ç.K. (eds.) CHES 2000. LNCS, vol. 1965, pp. 1–24. Springer, Heidelberg (2000)
Galbraith, S.: Supersingular curves in cryptography. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 495–513. Springer, Heidelberg (2001)
Verheul, E.R.: Evidence that XTR is more secure than supersingular elliptic curve cryptosystems. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 195–210. Springer, Heidelberg (2001)
Fong, K., Hankerson, D., López, J., Menezes, A.: Field inversion and point halving revisited, Technical Report CORR 2003-18, Univ. of Waterloo (2003)
Gao, S., von zur Gathen, J., Panario, D.: Gauss periods and fast exponentiation in finite fields. In: Baeza-Yates, R., Poblete, P.V., Goles, E. (eds.) LATIN 1995. LNCS, vol. 911, pp. 311–322. Springer, Heidelberg (1995)
Baek, J., Zheng, Y.: Identity-based threshold signature scheme from the bilinear pairings. In: ITCC 2004, Proceedings of International Conference on Information Technology, vol. 1, pp. 124–128 (2004)
Gaudry, P., Hess, F., Smart, N.P.: Constructive and destructive facets of Weil descent on elliptic curves. J. of Cryptology 15, 19–46 (2002)
Koblitz, N., Menezes, A., Vanstone, S.: The state of elliptic curve cryptography. Design, Codes and Cryptography 19, 173–193 (2000)
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Kwon, S. (2005). Efficient Tate Pairing Computation for Elliptic Curves over Binary Fields. In: Boyd, C., González Nieto, J.M. (eds) Information Security and Privacy. ACISP 2005. Lecture Notes in Computer Science, vol 3574. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11506157_12
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DOI: https://doi.org/10.1007/11506157_12
Publisher Name: Springer, Berlin, Heidelberg
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