Abstract
Hyperelliptic curves have been used to define discrete logarithm problems as cryptographic one-way functions. However, no efficient algorithm for construction of secure hyperelliptic curves is known until now. In this paper, efficient algorithms are presented to construct secure discrete logarithm problems on hyperelliptic curves whose Jacobian varieties are either simple or isogenous to a product of simple abelian varieties.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
L.M. Adleman, M.D.A. Huang: “Primality Testing and Abelian Varieties Over Finite Fields,” Springer-Verlag, (1992).
L.M. Adleman, J.D. Marrais, M.D. Huang: “A Subexponential Algorithms for Discrete Logarithms over the Rational Subgroup of the Jacobians of Large Genus Hyperelliptic Curves over Finite Fields,” Proc. of ANTS95, Springer, (1995)
D.Cantor: “Computing in the jacobian of hyperelliptic curve,” Math. Comp., vol.48, p.95–101, (1987)
J. Chao, K. Tanada, S. Tsujii: “Design of Elliptic Curves with Controllable Lower boundary of Extension Degree for Reduction Attacks”, Yvo G. Desmedt (Ed.) Advances in Cryptology-CRYPTO'94, Lecture Notes in Computer Science, 839, Springer-Verlag, pp.50–55, 1994.
J. Chao, K. Harada, N. Matsuda, S. Tsujii:“Design of secure elliptic curves over extension fields with CM fields methods,” Proc. of Pragocrypto'97, p.93–108, (1997)
M.D.Huang, D.Ierardi:“Counting Rational Point on Curves over Finite Fields,” Proc. 32nd IEEE Symp. on the Foundations of Computers Science, 1993.
K.Hashimoto, N.Murabayashi: “Shimura curves as intersections of Humbert surfaces and defining equations of QM-curves of genus two,” Tohoku Math.J. 47, p.271–296, (1995)
T.Honda: “Isogeny classes of abelian varieties over finite fields,” J.Math.Soc.Japan, vol.20, No.1–2, p.83–95, (1968)
J.Igusa: “Arithmetic variety of moduli for genus two,” Ann. of Math., vol.72, No.3, p.612–649, (1960)
N.Koblitz:“Elliptic Curve Cryptosystems,”Math. Comp.,vol.48, p.203–209, (1987)
N.Koblitz:“Hyperelliptic cryptosystems,” J. of Cryptology, vol.1, p.139–150, (1989)
N. Koblitz: “Elliptic Curve Implementation of Zero-Knowledge Blobs,” J. of Cryptology, vol.4, No.3, p. 207–213, (1991)
S.Lang: “Abelian Varieties”, Intersciencs, New York (1959)
S.Lang: “Complex multiplication” Springer-Verlag, (1983)
A.Menezes, S.Vanstone, T.Okamoto:“Reducing Elliptic Curve Logarithms to Logarithims in a Finite Fields,” Proc. of STOC, p.80–89, (1991).
A.Menezes:“Elliptic Curve Public Key Cryptosystems”, Kluwer Academic, (1993)
V.S.Miller: “Use of Elliptic Curves in Cryptography,” Advances in Cryptology Proceedings of Crypto'85, Lecture Notes in Computer Science, 218, Springer-Verlag, p.417–426, (1986)
D.Mumford: “Abelian varieties”, Tata Studies in Mathematics, Oxford, Bobay, (1970).
D.Mumford: “Tats Lectures on Theta I”, Birkhäuser, Boston, (1983).
D.Mumford: “Tata Lectures on Theta II”, Birkhäuser, Boston, (1984).
T.Okamoto, K.Sakurai: “Efficient Algorithms for the Construction of Hyperelliptic Cryptosystems,” Proc. of CRYPTO'91, LNCS 576, p.267–278, (1992).
J.Pila: “Frobenius maps of abelian varieties and finding roots of unity in finite fields,” Math. Comp., vol.55, p. 745–763, (1990)
R.Schoof: “Elliptic curves over finite fields and the computation of square roots mod p,” Math. Comp., vol.44, p.483–494, (1985)
G. Shimura, Y. Taniyama: “Complex multiplication of abelian varieties and its application to number theory” Pub. Math. Soc. Jap. no.6, (1961).
Emil J. Volcheck: “Computing in the Jacobian of a plane algebraic curve”, Proc. of ANT-1, p.221–233, LNCS-877, (1994)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag
About this paper
Cite this paper
Chao, J., Matsuda, N., Tsujii, S. (1997). Efficient construction of secure hyperelliptic discrete logarithm problems. In: Han, Y., Okamoto, T., Qing, S. (eds) Information and Communications Security. ICICS 1997. Lecture Notes in Computer Science, vol 1334. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0028485
Download citation
DOI: https://doi.org/10.1007/BFb0028485
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63696-0
Online ISBN: 978-3-540-69628-5
eBook Packages: Springer Book Archive