Abstract
In this paper we propose a key-exchange system and a public-key encryption scheme based on the class semigroups of imaginary quadratic non-maximal orders, the former is analogous to the Diffie-Hellman’s key-exchange system and the latter is similar to the ElGamal’s encryption scheme, whose security is based on the difficulty of the discrete logarithm problem of that class semigroup.
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References
Buchmann, J., Düllmann, S.: On the computation of discrete logarithm in class groups, in Advances in Cryptology — CRYPTO’ 90, LNCS 537, Springer-Velag, Berlin, 1991, pp. 134–139.
Buchmann, J., Hamdy, S.: A survey on IQ cryptography. Technical Report No. TI-4/01, Darmstadt University of Technology, 2001.
Buchmann, J., Paulus, S.: A one way function based on ideal arithmetic in number fields, in Advances in Cryptology — CRYPTO’ 97, LNCS 1294, Springer-Velag, Berlin, 1997, pp. 385–394.
Buchmann, J., Willams, H. C.: A key-exchange system based on imaginary quadratic fields. J. Cryptology 1 (1988) 107–118.
Cohen, H.: A course in Computational Algebraic Number Theory, Springer, Berlin, 1995.
Cox, D.: Primes of the Form x 2 + ny 2, Wiley, New York, 1989.
Delfs, H., Knebel, H.: Introduction to Cryptography: Principles and Applications, Springer-Verlag, Berlin, 2002.
Diffie, W., Hellman, M.: New directions in cryptography. IEEE Trans. Inform. Theory 22 (1976) 472–492.
ElGamal T.: A Public key cryptosystem and a signature scheme based on discrete logarithms. IEEE Trans. Inform. Theory 31 (1985), 469–472.
Hafner, J. L., McCurley, K. S.: A rigorous subexponential algorithm for computation of class group. J. Amer. Math. Soc. 2 (1989) 837–850.
Hamdy, S., Möller, B.: Security of cryptosystems based on class groups of imaginary quadratic orders, in Advances in Cryptology — ASIACRYPT 2000, LNCS 1976, Springer-Velag, Berlin, 2000, pp. 234–247.
Jacobson Jr., M. J.: Computing discrete logarithms in quadratic orders. J. Cryptology 13 (2000) 473–492.
Hühnlein, D., Takagi, T.: Reducing logarithms in totally non-maximal imaginary quadratic orders to logarithms in finite fields, in Advances in Cryptology — ASIACRYPT’ 99, LNCS 1716, Springer-Verlag, Berlin, 1999, pp. 219–231.
Koblitz, N.: Elliptic curve cryptosystems. Math. Comp. 48 (1987) 203–209.
Koblitz, N.: Hyperelliptic cryptosystems. J. Cryptology 1 (1989) 139–150.
McCurley, K. S.: Cryptographic key distribution and computation in class groups, in R. A. Mollin, editor, Number Theory and Applications, Kluwer Academic Publishers, 1989, pp. 459–479.
Menezes, A. J., Oorschot, P. C., Vanstone, S. A.: Handbook of Applied Cryptography, CRC Press, Boca Raton, 1997.
Meyer, A., Neis, S., Pfahler, T.: First implementation of cryptographic protocols based on algebraic number fields, in Information Security and Privacy, LNCS 2119, Springer-Velag, Berlin, 2001, pp. 84–103.
Mollin, R. A.: Quadratics, CRC Press, Boca Raton, 1996.
Odlyzko, A. M.: Discrete logarithms in finite fields and their cryptographic significance, Advances in Cryptology — EUROCRYPT’ 84, LNCS 209, Springer-Velag, Berlin, 1985, pp. 224–314.
Paulus, S., Takaki, T.: A new public-key cryptosystem over a quadratic order with quadratic decryption time. J. Cryptology 13 (2000) 263–272.
Rivest, R. L., Shamir, A., Adelman, L.: A method for abtaining digital signatures and public key cryptosystems. Communications of the ACM 21 (1978) 120–126.
Stinson, D. R.: Cryptography: Theory and Practice, CRC Press, Boca Raton, 2002.
Zanardo, P.: The class semigroup of orders in number fields. Math. Proc. Camb. Phil. Soc. 115 (1994) 379–391.
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Kim, H., Moon, S. (2003). Public-Key Cryptosystems Based on Class Semigroups of Imaginary Quadratic Non-maximal Orders. In: Safavi-Naini, R., Seberry, J. (eds) Information Security and Privacy. ACISP 2003. Lecture Notes in Computer Science, vol 2727. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45067-X_42
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DOI: https://doi.org/10.1007/3-540-45067-X_42
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