Abstract
In 2003, Kim and Moon [8] proposed two public-key cryptosystems based on arithmetic in the class semigroup of an imaginary non-maximal quadratic order. The authors argue that there is no known subexponential algorithm for solving the discrete logarithm problem in the class semigroup, and that as a result, their cryptosystems achieve a higher level of security as compared to those based on the class group. In this paper, we show that well-known structural properties of the class semigroup render these crytosystems insecure, and that any cryptosystems based on the class semigroup are unlikely to provide any more security than those using the class group.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Borevich, Z.I., Shafarevich, I.R.: Number theory. Academic Press, New York (1966)
Buchmann, J., Williams, H.C.: A key-exchange system based on imaginary quadratic fields. Journal of Cryptology 1, 107–118 (1988)
Cohen, H.: A course in computational algebraic number theory. Springer, Berlin (1993)
Cox, D.A.: Primes of the form x2 + ny2. John Wiley & Sons, New York (1989)
Hafner, J.L., McCurley, K.S.: A rigorous subexponential algorithm for computation of class groups. J. Amer. Math. Soc. 2, 837–850 (1989)
Hühnlein, D., Jacobson Jr., M.J., Weber, D.: Towards practical non-interactive public-key cryptosystems using non-maximal imaginary quadratic orders. Designs, Codes and Cryptography 30(3), 281–299 (2003)
Jacobson Jr., M.J., van der Poorten, A.J.: Computational aspects of NUCOMP. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 120–133. Springer, Heidelberg (2002)
Kim, H., Moon, S.: Public-key cryptosystems based on class semigroups of imaginary quadratic non-maximal orders. In: Safavi-Naini, R., Seberry, J. (eds.) ACISP 2003. LNCS, vol. 2727, pp. 488–497. Springer, Heidelberg (2003)
Neukirch, J.: Algebraische zahlentheorie. Springer, Berlin (1992)
Paulus, S., Takagi, T.: A new public-key cryptosystem over a quadratic order with quadratic decryption time. Journal of Cryptology 13, 263–272 (2000)
Zanardo, P., Zannier, U.: The class semigroup of orders in number fields. Math. Proc. Camb. Phil. Soc. 115, 379–391 (1994)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Jacobson, M.J. (2004). The Security of Cryptosystems Based on Class Semigroups of Imaginary Quadratic Non-maximal Orders. In: Wang, H., Pieprzyk, J., Varadharajan, V. (eds) Information Security and Privacy. ACISP 2004. Lecture Notes in Computer Science, vol 3108. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27800-9_13
Download citation
DOI: https://doi.org/10.1007/978-3-540-27800-9_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22379-5
Online ISBN: 978-3-540-27800-9
eBook Packages: Springer Book Archive