Abstract
This paper gives an efficient algorithm to compute addition in Jacobian of C34 curves. The paper modifies the addition algorithm of [1], by classifying the forms of Groebner bases of all ideals involved in the addition in Jacobian, and by computing Groebner bases of ideals without using Buchberger algorithm. The algorithm computes the addition in Jacobian of C34 curves in about 3 times amount of computation of the one in elliptic curves, when the sizes of groups are set to be the same.
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References
S. Arita, “Algorithms for computations in Jacobian group of C ab curve and their application to discrete-log-based public key cryptosystems,” IEICE TRANS., VOL.J82-A, NO.8, pp.1291–1299, 1999.
D.G. Cantor, “Computing in the Jacobian of a hyperelliptic curve”, Mathematics of Computation, 48(177), pp.95–101, 1987.
S.D. Galbraith, S. Paulus, and N.P. Smart “Arithmetic on Superelliptic Curves”, J. Cryptology (1999) 12, 193–196.
R. Harasawa, J. Suzuki, “A Fast Jacobian Group Arithmetic Scheme for Algebraic Curve Cryptography”, IEICE TRANS., Vol.E84-A No.1, pp.130–139, 2001
R. Harley, http://cristal.inria.fr/ harley/hyper/adding.text
R. Hartshorne, “Algebraic Geometry”, Springer-Verlag, 1977.
T. Lange, “Weighted Coordinates on Genus 2 Hyperelliptic Curves”, preprint.
R. Matsumoto, “The Cab Curve — a generalization of the Weierstrass form to arbitrary plane curves”, http://www.rmatsumoto.org/cab.html
K. Matsuo, J. Chao, “Fast Cryptosystems Using Genus 2 Hyperelliptic curves”, preprint.
S. Miura, “Linear Codes on Affine Algebraic Curves”, Trans. of IEICE, vol. J81-A, No. 10, 1398–1421, Oct. 1998.
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© 2003 Springer-Verlag Berlin Heidelberg
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Arita, S. (2003). An Addition Algorithm in Jacobian of C 34 Curve. In: Safavi-Naini, R., Seberry, J. (eds) Information Security and Privacy. ACISP 2003. Lecture Notes in Computer Science, vol 2727. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45067-X_9
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DOI: https://doi.org/10.1007/3-540-45067-X_9
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