• Communications of the Korean Mathematical Society
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• v.21 no.3
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• pp.577-586
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• 2006

In this paper we will propose the methods for finding the non-invertible ideals corresponding to non-primitive quadratic forms and clarify the structures of class SEMIGROUPS of imaginary quadratic orders which were given by Zanardo and Zannier [8], and we will give a general algorithm for calculating power of ideals/classes via the Dirichlet composition of quadratic forms which is applicable to cryptography in the class semigroup of imaginary quadratic non-maximal order and revisit the cryptosystem of Kim and Moon [5] using a Zanardo and Zannier [8]'s quantity as their secret key, in order to analyze Jacobson [7]'s revised cryptosystem based on the class semigroup which is an alternative of Kim and Moon [5]'s.

• Honam Mathematical Journal
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• v.26 no.3
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• pp.247-256
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• 2004

Buchmann and Williams[1] proposed a key exchange system making use of the properties of the maximal order of an imaginary quadratic field. $H{\ddot{u}}hnlein$ et al. [6,7] also introduced a cryptosystem with trapdoor decryption in the class group of the non-maximal imaginary quadratic order with prime conductor q. Their common techniques are based on the properties of the invertible ideals of the maximal or non-maximal orders respectively. Kim and Moon [8], however, proposed a key-exchange system and a public-key encryption scheme, based on the class semigroups of imaginary quadratic non-maximal orders. In Kim and Moon[8]'s cryptosystem, a non-invertible ideal is chosen as a generator of key-exchange ststem and their secret key is some characteristic value of the ideal on the basis of Zanardo et al.[9]'s quantity for ideal equivalence. In this paper we propose the methods for finding the non-invertible ideals corresponding to non-primitive quadratic forms and clarify the structure of the class semigroup of non-maximal order as finitely disjoint union of groups with some quantities correctly. And then we correct the misconceptions of Zanardo et al.[9] and analyze Kim and Moon[8]'s cryptosystem.

• Honam Mathematical Journal
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• v.28 no.2
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• pp.185-196
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• 2006

A scheme based on the cryptography for enforcing multilevel security in a system where hierarchy is represented by a partially ordered set was first introduced by Akl et al. But the key generation algorithm of Akl et al. is infeasible when there is a large number of users. In 1985, MacKinnon et al. proposed a paper containing a condition which prevents cooperative attacks and optimizes the assignment in order to overcome this shortage. In 2005, Kim et al. proposed key management systems for multilevel security using one-way hash function, RSA algorithm, Poset dimension and Clifford semigroup in the context of modern cryptography. In particular, the key management system using Clifford semigroup of imaginary quadratic non-maximal orders is based on the fact that the computation of a key ideal $K_0$ from an ideal $EK_0$ seems to be difficult unless E is equivalent to O. We, in this paper, show that computing preimages under the bonding homomorphism is not difficult, and that the multilevel cryptosystem based on the Clifford semigroup is insecure and improper to the key management system.