Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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ON THE STRUCTURES OF CLASS SEMIGROUPS OF QUADRATIC NON-MAXIMAL ORDERS

  • KIM, YONG TAE (Department of Mathematics Education, Gwangju National University of Education)
  • Received : 2004.06.24
  • Accepted : 2004.07.13
  • Published : 2004.09.25
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Abstract

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.

Keywords

Acknowledgement

Supported by : Gwangju National University of Education

References

  1. J. Cryptology v.1 A key exchange system based on imaginary quadratic fields Buchmann, J.;Willams, H.C.
  2. Primes of the form $x^2 \;+ \;ny^2$ Cox, D.
  3. Disquisitiones Arithmeticae Gauss, C.F.;Clarke A.A.(tr.)
  4. Multiplicative ideal theory Gilmer, R.
  5. An introduction to semigroup theory Howie, J.M.
  6. Advanced Cryptology Eurocrypt '98, LNCS 1403 A cryptosystem based on the non-maximal imaginary quadratic orders with fast decryption Huhnlein, D.;Michael, J.J. Jr.;Paulus, S.;Tagaki, T.
  7. Advances in Cryptology - ASIACRYPTO '99, LNCS 1716 Reducing logarithms in the totally non-maximal imaginary quadratic orders Huhnlein, D.;Tagaki, T.
  8. ASISP 2003, LNCS 2727 Public-Key Cryptosystems based on Class Semigroups of Imaginary Quadratic Non-maximal Orders Kim, H.;Moon, S.
  9. Math. Proc. Camb. Phil. Soc. v.115 The class semigroup of orders in number fields Zanardo, P.;Zannier, U.