Abstract
The goal of this paper is to describe a practical and efficient algorithm for computing in the Jacobian of a large class of algebraic curves over a finite field. For elliptic and hyperelliptic curves, there exists an algorithm for performing Jacobian group arithmetic in O(g 2) operations in the base field, where g is the genus of a curve. The main problem in this paper is whether there exists a method to perform the arithmetic in more general curves. Galbraith, Paulus, and Smart proposed an algorithm to complete the arithmetic in O(g 2) operations in the base field for the so-called superelliptic curves. We generalize the algorithm to the class of C ab curves, which includes superelliptic curves as a special case. Furthermore, in the case of C ab curves, we show that the proposed algorithm is not just general but more efficient than the previous algorithm as a parameter a in C ab curves grows large.
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References
Arita, S.: Algorithms for Computations in Jacobian Group of Cab Curve and Their Application to Discrete-Log Based Public Key Cryptosystems. IEICE Trans part A J82-A(8), 1291–1299 (1999) (in Japanese)
Cantor, D.G.: Computing in the Jacobian of a hyper-elliptic curves. Math.Comp. 48, 95–101 (1987)
Cohen, H.: A Course in Computational Algebraic Number Theory, GTM 138. Springer, Heidelberg (1993)
Frey, G., Rück, H.: A remark concerningm-divisibility and the discrete logarithm in the divisor class group of curves. Mathematics of Computation 62, 865–874 (1994)
Galbraith, S.D., Paulus, S., Smart, N.P.: Arithmetic on Superelliptic Curves (1998) (preprint)
Hartshorne, R.: Algebraic Geometry, GTM 52. Springer, Heidelberg (1977)
Koblitz, N.: Hyperelliptic cryptosystems. J. Cryptography 1, 139–150 (1989)
Miller, V.S.: Use of elliptic curves in cryptography, Advances in Cryptography CRYPTO 1985. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–426. Springer, Heidelberg (1986)
Miura, S.: The study of error coding codes based on algebraic geometry, Dr. thesis (1997) (in Japanese)
Müller, A.: Effiziente Algorithmen für Probleme der linearen Algebra über Z, Master’s thesis, Universität des Saarlandes, Saarbrücken (1994)
Paulus, S.: Lattice basis reduction in function field in Ants-3, Algorithmic Number Theory. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 567–575. Springer, Heidelberg (1998)
Paulus, S., Stein, A.: Comparing Real and Imaginary Arithmetics for Divisor Class Groups of Hyperelliptic Curves in Ants-3, Algorithmic Number Theory. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 576–591. Springer, Heidelberg (1998)
Silverman, J.H.: The Arithmetic of Elliptic Curves, Graduate Texts in Math., vol. 106. Springer, Heidelberg (1994)
Smart, N.P.: On the performance of Hyperelliptic Cryptosystems, Advances in Cryptology EUROCRYPTO 1999. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 165–175. Springer, Heidelberg (1999)
Stichtenoth, H.: Algebraic Function Fields and Codes. Springer, Heidelberg (1993)
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Harasawa, R., Suzuki, J. (2000). Fast Jacobian Group Arithmetic on C ab Curves. In: Bosma, W. (eds) Algorithmic Number Theory. ANTS 2000. Lecture Notes in Computer Science, vol 1838. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10722028_21
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DOI: https://doi.org/10.1007/10722028_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67695-9
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