Abstract
We investigate the discrete logarithm problem over jacobian varieties of hyperelliptic curves suitable for public-key cryptosystems, and clarify practical advantages of hyperelliptic cryptosystems compared to the elliptic cryptosystems and to RSA. We focus on the curves defined over the ground field of characteristic 2, and we present hyperelliptic cryptosystems from the jacobian associated with curves C : v 2 + v=u 2g+1 of genus g=3 and 11, which are secure against the known attacks. We further discuss the efficiency in implementation of such secure hyperelliptic cryptosystems.
A part of this work was done while visiting in Columbia Univ. Computer Science Dept. from September 1997 for one year.
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, J. DeMarrais and M. Huang, ”A Subexponential Algorithm for Discrete Logarithm over the Rational Subgroup of the Jacobians of Large Genus Hyperelliptic Curves over Finite Fields”, Proc. of ANTS1, LNCS, vol. 877, Springer-Verlag, (1994), 28–40
G.B. Agnew, R.C. Mullin and S.A. Vanstone, ”An Implementation of Elliptic Curve Cryptosystems Over \(F_{2^{155} }\)”, IEEE J. Selected Areas in Communications11, No.5 (1993), 804–813
T. Beth and F. Scaefer, ”Non supersingular elliptic curves for public key cryptosystems”, Advances in Cryptology — EUROCRYPT '91, Lecture Notes in Computer Science, 547, pp.316–327 (1991).
D.G. Cantor, ”Computing in the Jacobian of a Hyperelliptic Curve”, Math. Comp, 48, No.177 (1987), 95–101
J. Chao, K. Tanaka, and S. Tsujii, ”Design of elliptic curves with control-lable lower boundary of extension degree for reduction attacks”, Advances in Cryptology — Crypto'94, Springer-Verlag, (1994), 50–55.
G. Frey, ” Aspects of DL-systems based on hyperelliptic curves”, Keynote Lecture in Waterloo-Workshop on Elliptic Curve Discrete Logarithm Problem, 4th of Nov. (1997).
G. Frey and H.G. Rück, ”A Remark Concerning m-Divisibility and the Discrete Logarithm in the Divisor Class Group of Curves”, Math. Comp, 62, No.206 (1994), 865–874
T. Itoh, O. Teechai and S. Tsujii, ”A fast algorithm for computing multiplicative inverse in GF(2t) using normal bases” (in Japanese), J. Society for Electronic Communications (Japan), 44, (1986), 31–36.
D.E. Knuth, ”The Art of Computer Programing, Vol.2, Seminumerical Algo-rithm”, Addison-Wesley, Reading MA, 2nd edition (1981)
N. Koblitz, ”Elliptic curve cryptosystems”, Mathematics of Computation, 48 (1987), 203–209.
N. Koblitz, ”A Family of Jacobians Suitable for Discrete Log Cryptosystems”, Advances in Cryptology — Crypto'88, Springer-Verlag, (1990), 94–99
N. Koblitz, ”Hyperelliptic Cryptosystems”, J.Cryptology, 1 (1989), 139–150
N. Koblitz, ”A Very Easy Way to Generate Curves over Prime Fields for Hyperelliptic Cryptosystems”, Crypto'97 Rump Talk (1997)
V. Miller, ”Uses of elliptic curves in cryptography”, Lecture Notes in Computer Science, 218 (1986), 417–426. (Advances in Cryptology — CRYPTO '85.)
K. Matsuo, J. Chao and S.Tsujii, ”Design of Cryptosystems Based on Abelian Varieties over Extension Fields”, IEICE ISEC, 97–30 (1997), 9–18
A. Miyaji, ”Elliptic curve over Fp suitable for cryptosystems”, Advances in Cryptology — Asiacrypt'92, Springer-Verlag, (1993), 479–491.
A. Miyaji, ”Elliptic curve cryptosystems immune to any reduction into the discrete logarithm problem”, IEICE Trans., Fundamentals, E76-A (1993), pp. 50–54.
A.J. Menezes, T. Okamoto and S.A. Vanstone, ”Reducing elliptic curve logarithm to logarithm in a finite field”, IEEE Trans. on IT, 39, (1993), 1639–1646
R.C. Mullin, I.M. Onyszchuk, S.A. Vanstone and R.M. Wilson,”Optimal Normal Bases in GF(pn)”, Discrete Applied Mathematics, 22, (1988/89), 149–161
A. Odlyzko, ”Discrete logarithm and their cryptographic significance”, Advances in Cryptology — Eurocrypto'84, Springer-Verlag, (1985), 224–314
J. Pila, ”Frobenius maps of abelian varieties and finding roots of unity in finite fields”, Math. Comp, 55, No.206 (1990), 745–763.
S.C. Pohlig and M.E. Hellman, ”An improved algorithm for computing logarithms over GF(p) and its cryptographic significance”, IEEE Trans. on IT, 24, (1978), 106–110
R. Lidl and H. Niederreiter, ”Finite Fields”, Encyclopedia of Mathematics and Its Application, (1987)
http://www.rsa.com
H.G. Rück, ”On the discrete logarithms in the divisor class group of curves”, To appear in Math. Comp. (1997)
T. Satoh and K. Araki, ”Fermat Quotients and the Polynomial Time Discrete Log Algorithm for Anomalous Elliptic Curves”, preprint, (1997)
I.A. Semaev, ”Evaluation of discrete logarithms in a group of p-torsion points of an elliptic curve in characteristic p”, Math. Comp., Vol.76 (1998),pp.353–356.
R. Schoof, ”Elliptic curves over finite fields and the computation of square root mod p”, Math. Comp, 44, (1985), 483–494.
N.P. Smart, ”The Discrete Logarithm Problem on Elliptic Curves of Trace One”, preprint, (1997)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sakai, Y., Sakurai, K., Ishizuka, H. (1998). Secure hyperelliptic cryptosystems and their performance. In: Imai, H., Zheng, Y. (eds) Public Key Cryptography. PKC 1998. Lecture Notes in Computer Science, vol 1431. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0054023
Download citation
DOI: https://doi.org/10.1007/BFb0054023
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-64693-8
Online ISBN: 978-3-540-69105-1
eBook Packages: Springer Book Archive