Abstract
In this paper, we first introduce a shift product-based polynomial transformation. Then, we show that the parities of (#E – 1)/2 and (#E′ – 1)/2 are reciprocal to each other, where #E and #E′ are the orders of the two candidate curves obtained at the last step of CM method algorithm. Based on this property, we propose a method to check the parity by using the shift product-based polynomial transformation. For a 160-bits prime number as the characteristic, the proposed method carries out the parity check about 20 times faster than the conventional method when 4 divides the characteristic minus 1.
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References
Sato, T., Araki, K.: Fermat Quotients and the Polynomial Time Discrete Lot Algorithm for Anomalous Elliptic Curve. Commentarii Math. Univ. Sancti. Pauli 47(1), 81–92 (1998)
Frey, G., Rück, H.: A Remark Concerning m-Divisibility and the Discrete Logarithm in the Divisor Class Group of Curves. Math. Comp. 62, 865–874 (1994)
Gaudry, P., Hess, F., Smart, N.: Constructive and destructive facets of Weil descent on elliptic curves. Hewlett Packard Tech. Report HPL-2000-10 (2000)
Horiuchi, K., et al.: Construction of Elliptic Curves with Prime Order and Estimation of Its Complexity. IEICE Trans. J82-A(8), 1269-1277 (1999)
Konstantinou, E., Stamatiou, Y.C., Zaroliagis, C.D.: On the construction of prime order elliptic curves. In: Johansson, T., Maitra, S. (eds.) INDOCRYPT 2003. LNCS, vol. 2904, pp. 309–322. Springer, Heidelberg (2003)
Nogami, Y., Morikawa, Y.: Fast Generation of Elliptic Curves with Prime Order over \(F_{p^{2^c}}\). In: Proc. of Workshop on Coding and Cryptography, pp. 347–356 (2003)
Savas, E., Schmidt, T., Koc, C.: Generating Elliptic Curves of Prime Order. In: Koç, Ç.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 142–158. Springer, Heidelberg (2001)
Blake, I., Seroussi, G., Smart, N.: Elliptic Curves in Cryptography. LNS, vol. 265. Cambridge University Press, Cambridge (1999)
Class polynomials of CM-fields, http://www.exp-math.uni-essen.de/zahlentheorie/classpol/class.html
Hiramoto, T., Nogami, Y., Morikawa, Y.: A Fast Algorithm to Test Irreducibility of Cubic Polynomial over GF(P). IEICE Trans. J84-A(5) (2000)
A Library for doing Number Theory, http://www.shoup.net/ntl/
Lidl, R., Niederreiter, H.: Finite Fields, Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (1984)
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Nogami, Y., Morikawa, Y. (2005). A Method for Distinguishing the Two Candidate Elliptic Curves in CM Method. In: Park, Cs., Chee, S. (eds) Information Security and Cryptology – ICISC 2004. ICISC 2004. Lecture Notes in Computer Science, vol 3506. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11496618_19
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DOI: https://doi.org/10.1007/11496618_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-26226-8
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