Abstract
Recently, the well-known Diffie-Hellman key exchange protocol was extended to real quadratic congruence function fields in a non-group based setting. Here, the underlying key space was the set of reduced principal ideals. This set does not possess a group structure, but instead exhibits a so-called infrastructure. The techniques are the same as in the protocol based on real quadratic number fields. As always, the security of the protocol depends on a certain discrete logarithm problem (DLP). It can be shown that for elliptic congruence function fields this DLP is equivalent to the DLP for elliptic curves over finite fields. In this paper, we present the arithmetic of reduced principal ideals in elliptic congruence function fields, which is the base for the equivalence, and prove some properties which have no analogies for real quadratic number fields.
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References
Abel, C. S.: Ein Algorithmus zur Berechnung der Klassenzahl und des Regulators reell-quadratischer Ordnungen. Dissertation, Universität des Saarlandes, Saarbrücken (Germany) 1994.
Artin, E.: Quadratische Körper im Gebiete der höheren Kongruenzen I, II. Math. Zeitschr. 19 (1924) 153–206.
Cohen, H.: A Course in Computation Algebraic Number Theory. Springer, Berlin 1994.
Deuring, M.: Lectures on the Theory of Algebraic Functions of One Variable. LNM 314, Berlin 1973.
Diffie, W., Hellman, M. E.: New Directions in Cryptography. IEEE Trans. Inform. Theory 22 no. 6 (1976) 644–654.
Scheidler, R., Buchmann, J. A., Williams, H. C. A key exchange protocol using real quadratic fields. J. Cryptology 7 (1994) 171–199.
Scheidler, R., Stein, A., Williams, H. C.: Key-exchange in real quadratic congruence function fields. Designs, Codes and Cryptology 7:1/2 (1996) 153–174.
Schmidt, F. K.: Analytische Zahlentheorie in Körpern der Charakteristik p. Math. Zeitschr. 33 (1931) 1–32.
Shanks, D.: The Infrastructure of a Real Quadratic Field and its Applications. Proc. 1972 Number Theory Conf., Boulder, Colorado (1972), 217–224.
Stein, A., Zimmer, H. G.: An Algorithm for Determining the Regulator and the Fundamental Unit of a Hyperelliptic Congruence Function Field. Proc. 1991 Int. Symp. on Symbolic and Algebraic Computation, Bonn (Germany), July 15–17, ACM Press, 183–184.
Stein, A.: Baby step-Giant step-Verfahren in reell-quadratischen Kongruenzfunktionenkörpern mit Charakteristik ungleich 2. Diplomarbeit, Universität des Saarlandes, Saarbrücken (Germany) 1992.
Stein, A.: Equivalences between Elliptic Curves and Real Quadratic Congruence Function Fields. submitted.
Stein, A., Williams, H. C.: Baby step giant step in real quadratic function fields. submitted.
Weis, B., Zimmer, H. G.: Artin's Theorie der quadratischen Kongruenzfunktionenkörper und ihre Anwendung auf die Berechnung der Einheiten-und Klassengruppen. Mitt. Math. Ges. Hamburg, Sond. XII (1991) no. 2.
Weiss, E.: Algebraic Number Theory. McGraw-Hill, New York 1963.
Williams, H. C., Wunderlich, M. C.: On the Parallel Generation of the Residues for the Continued Fraction Algorithm. Math. Comp. 48 (1987) 405–423.
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© 1996 Springer-Verlag Berlin Heidelberg
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Stein, A. (1996). Elliptic congruence function fields. In: Cohen, H. (eds) Algorithmic Number Theory. ANTS 1996. Lecture Notes in Computer Science, vol 1122. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61581-4_68
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DOI: https://doi.org/10.1007/3-540-61581-4_68
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