Abstract
We present a key exchange scheme similar to that of Diffie and Hellman using the infrastructure of quadratic function fields of even characteristic. This is a modification of the results of Scheidler, Stein and Williams who used quadratic function fields of odd characteristic. We also extend these results to give a digital signature scheme similar to that of ElGamal. These schemes are possible in this structure even though it is not a group. Finally we examine the security of such systems, and give a possible attack based on Pohlig and Hellman's attack on discrete logarithms in finite groups.
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References
W. Diffie and M.E. Hellman, New directions in cryptography, IEEE Transactions on Information Theory, Vol. 22 (1976) pp. 644–654.
T. ElGamal, A public key cryptosystem and a signature scheme based on discrete logarithms, IEEE Transactions on Information Theory, Vol. 31 (1985) pp. 469–472.
V. Müller, A. Stein and C. Thiel, Computing discrete logarithms in real quadratic congruence function fields of large genus, preprint.
R. Mullin, I. Onyszchuk, S. Vanstone, and R. Wilson, Optimal normal bases in GF(p n), Discrete Applied MathematicsVol. 22 (1988/1989) pp. 149–161.
S. Pohlig and M. Hellman, An improved algorithm for computing logarithms over GF(p) and its cryptographic significance, IEEE Transactions on Information Theory, Vol. 24 (1978) pp. 918–924.
R. Scheidler, Cryptography in real quadratic congruence function fields, Proceedings of Pragocrypt 1996, CTU Publishing House, Prague, Czech Republic (1996).
R. Scheidler, J.A. Buchmann and H.C. Williams, Akey exchange protocol using real quadratic fields, Journal of Cryptology, Vol. 7 (1994) pp. 171–199.
R. Scheidler, A. Stein and H.C. Williams, Key-exchange in real quadratic congruence function fields, Designs, Codes and Cryptography, Vol. 7 (1996) pp. 153–174.
F.K. Schmidt, Analytische Zahlentheorie in Körpern der Charakteristik p, Mathematische Zeitschrift, Vol. 33 (1931), pp. 1–32.
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 (1992) Saarbrücken.
A. Stein, Equivalences between elliptic curves and real quadratic congruence function fields, to appear in Jorunal de Theorie des Nombres de Bordeaux(1997).
A. Stein and H.C. Williams, Baby step-giant step in real quadratic congruence function fields, preprint.
H. Stichtenoth, Algebraic Function Fields and Codes, Springer-Verlag, Berlin (1993).
B. Weiss and H.G. Zimmer, Artin's Theorie der quadratischen Kongruenzfunkionenkörper und ihre Anwendung auf die Berechnung der Einheiten-und Klassengrupen, Mitteilungen der Mathematischen Gesellschaft in Hamburg, Vol. XII (1991) pp. 261–286.
R.J. Zuccherato, The continued fraction algorithm and regulator for quadratic function fields of characteristic 2, Journal of Algebra, Vol. 190 (1997) pp. 563–587.
R.J. Zuccherato, New Applications of Elliptic Curves and Function Fields in Cryptography, Ph.D. Thesis (1997) Department of Combinatorics and Optimization, University of Waterloo.
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Müller, V., Vanstone, S. & Zuccherato, R. Discrete Logarithm Based Cryptosystems in Quadratic Function Fields of Characteristic 2. Designs, Codes and Cryptography 14, 159–178 (1998). https://doi.org/10.1023/A:1008240113843
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DOI: https://doi.org/10.1023/A:1008240113843