Abstract
We show how the theory of real quadratic congruence function fields can be used to produce a secure key distribution protocol. The technique is similar to that advocated by Diffie and Hellman in 1976, but instead of making use of a group for its underlying structure, makes use of a structure which is “almost” a group. The method is an extension of the recent ideas of Scheidler, Buchmann and Williams, but, because it is implemented in these function fields, several of the difficulties with their protocol can be eliminated. A detailed description of the protocol is provided, together with a discussion of the algorithms needed to effect it.
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References
C. S. Abel, Ein Algorithmus zur Berechnung der Klassenzahl und des Regulators reellquadratischer Ordnungen, Dissertation, Universität des Saarlandes, Saarbrücken (1994).
G. B. Agnew, R. C. Mullin and S. A. Vanstone, An implementation of elliptic curve cryptosystems over % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFfcVrdaWgaaWcbaGa% aGOmamaaCaaameqabaGaaGymaiaaiwdacaaI1aaaaaWcbeaaaaa!4529!\[\mathbb{F}_{2^{155} } \], IEEE J. Selected Areas in Communications, Vol. 11 (1993) pp. 804–813.
E. Artin, Quadratische Körper im Gebiete der höheren Kongruenzen I, II, Math. Zeitschr., Vol. 19 (1924) pp. 153–206.
H. Cohen, A Course in Computation Algebraic Number Theory, Springer, Berlin (1994).
H. Cohen and H. W. Lenstra, Heuristics on class groups, in Number Theory (H. Jager, ed.) (Noordwijkerhout, 1983), Lecture Notes in Mathematics, Springer, New York, 1052 (1984) pp. 26–36.
H. Cohen and H. W. Lenstra, Heuristics on class groups of number fields, in Number Theory (H. Jager, ed.) (Noordwijkerhout, 1983), Lecture Notes in Mathematics, Springer, New York, 1068 (1984) pp. 33–62.
M. Deuring, Lectures on the Theory of Algebraic Functions of One Variable, Lecture Notes in Mathematics, Berlin 314 (1973).
W. Diffie and M. E. Hellman, New directions in cryptography, IEEE Trans. Inform. Theory, Vol. 22, No. 6, (1976) pp. 644–654.
M. Eichler, Introduction to the Theory of Algebraic Numbers and Functions, Academic Press, New York (1966).
E. Friedman and L. C. Washington, On the distribution of divisor class groups of curves over finite fields, Theorie des Nombres, Proc. Int. Number Theory Conf. Laval, 1987, Walter de Gruyter, Berlin and New York (1989) pp. 227–239.
H. W. Lenstra Jr., On the calculation of regulators and class numbers of quadratic fields, London Math. Soc. Lec. Note Ser., Vol. 56, (1982) pp. 123–150.
R. Scheidler, J. A. Buchmann and H. C. Williams, A key exchange protocol using real quadratic fields, J. Cryptology, Vol. 7, (1994) pp. 171–199.
F. K. Schmidt, Analytische Zahlentheorie in Körpern der Charakteristik p, Math. Zeitschr., Vol. 33 (1931) pp. 1–32.
R. J. Schoof, Quadratic fields and factorization, Computational Methods in Number Theory (H. W. Lenstra and R. Tijdemans, eds.), Math. Centrum Tracts, Part II, Amsterdam, 155 (1983) pp. 235–286.
D. Shanks, The infrastructure of a real quadratic field and its applications, Proc. 1972 Number Theory Conf., Boulder, Colorado (1972) pp. 217–224.
A. Stein, Baby step-Giant step-Verfahren in reell-quadratischen Kongruenzfunktionenkörpern mit Charakteristik ungleich 2, Diplomarbeit, Universität des Saarlandes, Saarbrücken (1992).
A. Stein, Equivalences between Elliptic Curves and Real Quadratic Congruence Function Fields, in preparation.
A. Stein and H. G. Zimmer, 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, ACM Press, July 15–17 (1991) pp. 183–184.
B. Weis and H. G. Zimmer, Artin's Theorie der quadratischen Kongruenzfunktionenkörper und ihre Anwendung auf die Berechnung der Einheiten-und Klassengruppen, Mitt. Math. Ges. Hamburg, Sond., Vol. XII, No. 2 (1991) pp. 261–286.
E. Weiss, Algebraic Number Theory, McGraw-Hill, New York (1963).
X. Zhang, Ambiguous classes and 2-rank of class groups of quadratic function fields, J. of China University of Science and Technology, Vol. 17, No. 4, (1987) pp. 425–431.
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Dedicated to Gus Simmons
Research supported by NSERC of Canada Drant #A7649
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Scheidler, R., Stein, A. & Williams, H.C. Key-exchange in real quadratic congruence function fields. Des Codes Crypt 7, 153–174 (1996). https://doi.org/10.1007/BF00125081
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DOI: https://doi.org/10.1007/BF00125081