The security of cryptosystems based on class semigroups of imaginary quadratic non-maximal orders

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. © Springer-Verlag Berlin Heidelberg 2004.