Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ON THE PUBLIC KEY CRYPTOSYSTEMS OVER CLASS SEMIGROUPS OF IMAGINARY QUADRATIC NON-MAXIMAL ORDERS

  • Kim, Young-Tae (Department of Mathemathics Education Gwangju National University of Education) ;
  • Kim, Chang-Han (Department of Information Security Semyung University)
  • Published : 2006.07.01
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Abstract

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.

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References

  1. J. Buchmann and H. C. Williams, A key-exchange system based on imaginary quadratic fields, J. Cryptology 1 (1988), 107-118 https://doi.org/10.1007/BF02351719
  2. D. Cox, Primes of the form $x^2\;+\;ny^2$, Wiley, New York, 1989
  3. W. Diffie and M. Hellman, New directions in cryptography, IEEE Trans. on Inform. Theory 22 (1976), 472-492
  4. C. F. Gauss, Disquisitiones Arithmeticae , translated by Clarke A. A., SpringerVerlag, New York, 1986
  5. H. Kim and S. Moon, Public-Key Cryptosystems based on Class Semigroups of Imaginary Quadratic Non-maximal Orders, ASISP 2003, LNCS 2727 (2003), 488-497 https://doi.org/10.1007/3-540-45067-X_42
  6. Y. Kim, On the structures of class semiqroups of quadratic non-maximal orders, Honam Mathematical Journal 26 (2004), no. 3, 247-256
  7. Michael J. Jacobson, Jr., The security of cryptosystems based on class semigroups of imaginary quadratic non-maximal orders ASISP 2004, LNCS 3108 (2004), 149-156
  8. P. Zanardo and U. Zannier, The class semigroup of orders in number fields, Math. Proc. Camb.Phil. Soc. 115 (1994), 379-391