Cryptography in Quadratic Function Fields

Designs, Codes, and Cryptography

Volumn 22, Issue 3, 2001, Pages 239-264

  Scheidler, R ^a  

^a UNIVERSITY OF DELAWARE   (United States)

Author keywords

Diffie Hellman key exchange protocol; Discrete logarithm; ElGamal signature scheme; Public key cryptosystem; Quadratic function field

Indexed keywords

EID: 0035313617     PISSN: 09251022     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1008346322837     Document Type: Article

References (42)

1. C. Abel, Ein Algorithmus zur Berechnung der Klassenzahl und des Regulators reellquadratischer Ordnungen, Doctoral Dissertation, Universität des Saarlandes, Saarbrücken, Germany (1994).
2. L. M. Adleman and J. DeMarrais, A subexponential algorithm for discrete logarithms over all finite fields, Math. Comp., Vol. 61 (1993) pp. 1-15.
3. L. M. Adleman and M. D. Huang, Counting rational points on curves and Abelian varieties over finite fields, Second Internat. Symp. Algorithmic Number Theory ANTS-II, Lecture Notes in Computer Science, Springer, Berlin, 1122 (1996) pp. 1-16.
4. L. M. Adleman, J. DeMarrais and M. D. Huang, A subexponential algorithm for discrete logarithms over the rational subgroup of the Jacobians of large genus hyperelliptic curves over finite fields, First Internat. Symp. Algorithmic Number Theory ANTS-I, Lecture Notes in Computer Science Springer, Berlin, 877 (1994) pp. 28-40.
5. 2155, IEEE J. Selected Areas in Communications, Vol. 11 (1993) pp. 804-813.
6. E. Artin, Quadratische Körper im Gebiete der höheren Kongruenzen I, II, Math. Zeitschr., Vol. 19 (1924) pp. 153-206.
7. I. Biehl, B. Meyer and C. Thiel, Cryptographic protocols based on real-quadratic A-fields (extended abstract), Advances in Cryptology - ASIACRYPT 96, Lecture Notes in Computer Science, Springer, Berlin, 1163 (1996) pp. 15-25.
8. I. Biehl, J. A. Buchmann and C. Thiel, Cryptographic protocols based on discrete logarithms in real-quadratic orders, Proceedings of CRYPTO '94, Lecture Notes in Computer Science, Springer, New York, 839 (1994) pp. 56-60.
9. J. A. Buchmann, S. Düllmann and H. C. Williams, On the complexity of a new key exchange system, EUROCRYPT 90 Proceedings, Lecture Notes in Computer Science, Springer, Berlin, 473 (1990) 597-616.
10. J. A. Buchmann and H. C. Williams, A key-exchange system based on imaginary quadratic fields, J. Cryptology,Vol. 1 (1988) pp. 107-118.
11. J. A. Buchmann and H. C. Williams, A key-exchange system based on real quadratic fields, CRYPTO '89 Proceedings, Lecture Notes in Computer Science, Springer, Berlin 435 (1989) 335-343.
12. D. G. Cantor, Computing in the Jacobian of a hyperelliptic curve, Math. Comp., Vol. 48 (1987) pp. 95-101.
13. H. Cohen, A Course in Computational Algebraic Number Theory, Springer, Berlin 1995.
14. H. Cohen and H. W. Lenstra, Heuristics on class groups, Number Theory (H. Jager, ed.), Nordwijkerhout, (1983), Lecture Notes in Mathematics, Springer, New York, 1052 (1984) pp. 26-36.
15. H. Cohen and H. W. Lenstra, Heuristics on class groups of number fields, Number Theory (H. Jager, ed.) (Nordwijkerhout 1983), Lecture Notes in Mathematics, Springer, New York, 1068 (1984) pp. 33-62.
16. W. Diffie and M. E. Hellman, New directions in cryptography, IEEE Trans. Inform. Theory, Vol. 22 (1976) pp. 644-654.
17. T. ElGamal, A public-key cryptosystem and a signature scheme based on discrete logarithms, IEEE Trans. Inf. Theory, Vol. IT-31 (1985) pp. 469-472.
18. A. Enge, Computing discrete logarithms in high-genus hyperelliptic Jacobians in provably subexponential time. Research Report CORR # 99-04, University of Waterloo, Feb. 1999.
19. G. Frey and H. Rück, A remark concerning m-divisibility and the discrete logarithm in the divisor class group of curves, Math. Comp., Vol. 62 (1994) pp. 865-874.
20. C. Friesen, A special case of Cohen-Lenstra heuristics in function fields, Centre de Recherches Mathématique Proceedings and Lecture Notes, 19 (1999) pp. 99-105.
21. C. Friesen, Class group frequencies of real quadratic function fields: the degree 4 case. Math. Comp., Vol. 69 (2000) pp. 1213-1228.
22. C. Friesen, Bounds for frequencies of class groups of real quadratic genus 1 function fields. Preprint.
23. J. L. Hafner and K. S. McCurley, A rigorous subexponential algorithm for computation of class group, J. Amer. Math. Soc., Vol. 2 (1989) pp. 837-849.
24. E. Jung, Theorie der Algebraischen Funktionen einer Veränderlichen, Berlin (1923).
25. N. Koblitz, Hyperelliptic cryptosystems, J. Cryptology, Vol. 1 (1989) pp. 139-150.
26. A. Menezes, T. Okamoto and S. Vanstone, Reducing elliptic curve logarithms to logarithms in a finite field, IEEE Trans. Inf. Theory, Vol. 39 (1993) pp. 1639-1646.
27. V. Müller, A. Stein and C. Thiel, Computing discrete logarithms in real quadratic congruence function fields of large genus, Math. Comp., Vol. 68 (1999) pp. 807-822.
28. V. Müller, S. Vanstone and R. Zuccherato, Discrete Logarithm based cryptosystems in quadratic function fields of characteristic 2, Designs, Codes and Cryptography, Vol. 14 (1998) pp. 159-178.
29. National Institute for Standards and Technology, Digital Signature Standard, FIPS Publication, Vol. 186 (1993).
30. S. Paulus and H.-G. Rück, Real and imaginary quadratic representations of hyperelliptic function fields, Math. Comp., Vol. 68 (1999) pp. 1233-1241.
31. S. Paulus and A. Stein, Comparing real and imaginary arithmetics for divisor class groups of hyperelliptic curves, Third Internat. Symp. Algorithmic Number Theory ANTS-III, Lecture Notes in Computer Science, Springer, Berlin, 1423 (1998) pp. 576-591.
32. S. Pohlig and M. Hellman, An improved algorithm for computing logarithms over GF(p) and its cryptographic significance, IEEE Trans. Inf. Theory, Vol. 24 (1978) pp. 918-924.
33. R. Scheidler, Cryptography in real quadratic congruence function fields, Proceedings of PRAGOCRYPT 1996, CTU Publishing House, Prague, Czech Republic (1996) pp. 109-128.
34. 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.
35. 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.
36. D. Shanks, The infrastructure of a real quadratic field and its applications, Proc. 1972 Number Theory Conf., Boulder, Colorado, (1972) pp. 217-224.
37. A. Stein, Baby step-Giant step-Verfahren in reell-quadratischen Kongruenzfunktionenkörpern mit Charakteristik ungleich 2, Diplomarbeit, Universität des Saarlandes, Saarbrücken, Germany (1992).
38. A. Stein, Algorithmen in reell-quadratischen Kongruenzfunktionenkörpern, Doctoral Dissertation, Universität des Saarlandes, Saarbrücken, Germany (1996).
39. A. Stein, Equivalences between elliptic curves and real quadratic congruence function fields, J. Théorie des Nombres de Bordeaux, Vol. 9 (1997) pp. 75-95.
40. H. Stichtenoth, Algebraic Function Fields and Codes, Springer, Berlin (1993).
41. B. Weis and H. G. Zimmer, Artins Theorie der quadratischen Kongruenzfunktionenkörper und ihre Anwendung auf die Berechnung der Einheiten- und Klassengruppen, Mitt. Math. Ges. Hamburg, Vol. XII (1991) pp. 261-286.
42. X. Zhang, Ambiguous classes and 2-rank of class groups of quadratic function fields, J. China Univ. Sci. Tech., Vol. 17 (1987) pp. 425-431.