Computing discrete logarithms in quadratic orders

Journal of Cryptology

Volumn 13, Issue 4, 2000, Pages 473-492

  Jacobson Jr , Michael J ^a  

^a UNIVERSITY OF WATERLOO   (Canada)

Author keywords

, class group; Computational number theory; Discrete logarithm; Principal ideal testing; Quadratic order

Indexed keywords

EID: 0242324505     PISSN: 09332790     EISSN: None     Source Type: Journal    
DOI: 10.1007/s001450010013     Document Type: Article

References (32)

1. C.S. Abel, Ein Algorithmus zur Berechnung der Klassenzahl und des Regulators reellquadratischer Ordnungen, Ph.D. thesis, Universitat des Saarlandes, Saarbrucken, 1994.
2. I. Biehl, J. Buchmann, and C. Thiel, Cryptographic protocols based on discrete logarithms in real-quadratic orders, Advances in Cryptology - CRYPTO '94, Lecture Notes in Computer Science, vol. 839, Springer-Verlag, Berlin, 1995, pp. 56-60.
3. J. Buchmann, A subexponential algorithm for the determination of class groups and regulators of algebraic number fields, Séminaire de Théorie des Nombres (Paris), 1988-89, pp. 27-41.
4. J. Buchmann and S. Diillmann, On the computation of discrete logarithms in class groups, Advances in Cryptology - CRYPTO '90, Lecture Notes in Computer Science, vol. 537, Springer-Verlag, Berlin, 1991, pp. 134-139.
5. J. Buchmann, S. Düllmann, and H.C. Williams, On the complexity and efficiency of a new key exchange system, Advances in Cryptology - EUROCRYPT '89, Lecture Notes in Computer Science, vol. 434, Springer-Verlag, Berlin, 1990, pp. 597-616.
6. J. Buchmann, M.J. Jacobson, Jr., and E. Teske, On some computational problems in finite abelian groups, Math. Comp. 66 (1997), 1663-1687.
7. J. Buchmann and M. Maurer, Approximate Evaluation of L(1, χΔ), Tech. Report TI-6/98, Department of Computer Science, Technical University of Darmstadt, Darmstadt, 1998.
8. J. Buchmann, C. Thiel, and H.C. Williams, Short Representation of Quadratic Integers, Computational Algebra and Number Theory, Mathematics and its Applications, 325, Kluwer, Dordrecht, 1995, pp. 159-185.
9. J. Buchmann and H.C. Williams, A key-exchange system based on imaginary quadratic fields, J. Cryptology 1(1988), 107-118.
10. H. Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag, Berlin, 1993.
11. H. Cohen, F. Diaz y Diaz, and M. Olivier, Calculs de nombres de classes et de régulateurs de corps quadratiques en temps sous-exponentiel, Séminaire de Théorie des Nombres (Paris), 1993, pp. 35-46.
12. H. Cohen, F. Diaz y Diaz, and M. Olivier, Subexponential algorithms for class and unit group computations, J. Symbolic Comp. 24 (1997), 433-441.
13. H. Cohen and H.W. Lenstra, Jr., Heuristics on class groups, in Number Theory (Noordwijkerhout, 1983), Lecture Notes in Mathematics, vol. 1052, Springer-Verlag, New York, 1984, pp. 26-36.
14. S. Düllmann, Ein Algorithmus zur Bestimmung der Klassengruppe positiv definiter binarer quadratischer Formen, Ph.D. thesis, Universitat des Saarlandes, Saarbrucken, 1991.
15. T. El Gamal, A public key cryptoSystem and a signature scheme based on discrete logarithms, IEEE Trans. Inform. Theory 31 (1985), 469-472.
16. J.L. Hafner and K.S. McCurley, A rigorous subexpoential algorithm for computation of class groups, J. Amer. Math. Soc. 2 (1989), 837-850.
17. L.K. Hua, Introduction to Number Theory, Springer-Verlag, New York, 1982.
18. D. Huhnlein, MJ. Jacobson, Jr., S. Paulus, and T. Takagi, A cryptosystem based on non-maximal imaginary quadratic orders with fast decryption, Advances in Cryptology - EUROCRYPT '98, Lecture Notes in Computer Science, vol. 1403, Springer-Verlag, Berlin, 1998, pp. 294-307.
19. MJ. Jacobson, Jr., Subexponential Class Group Computation in Quadratic Orders, Ph.D. thesis, Technische Universitat Darmstadt, Darmstadt, 1999.
20. N. Koblitz, Elliptic curve cryptosystems, Math. Comp. 48 (1987), 203-209.
21. N. Koblitz, Hyperelliptic cryptosystems, J. Cryptology 1 (1989), 139-150.
22. H.W. Lenstra, Jr., On the Calculation of Regulators and Class Numbers of Quadratic Fields, London Mathematical Society Lecture Note Series, vol. 56, Cambridge University Press, Cambridge, 1982, pp. 123-150.
23. LiDIA, http://www.informatik.tu-darmstadt.de/TI/LiDIA, 1997.
24. M. Maurer, Regulator Approximation and Fundamental Unit Computation for Real-quadratic Orders, Ph.D. thesis, Technische Universitat Darmstadt, Darmstadt, 2000, in preparation.
25. K.S. McCurley, Cryptographic Key Distribution and Computation in Class Groups, NATO ASI on Number Theory and Applications (R.A. Mollin, ed.), Kluwer, Dordrecht, 1989, pp. 459-479.
26. A. Meyer, Ein neues Identifikations- und Signaturverfahren uber imaginar-quadratischen Zahlkorpern, Master's thesis, Universitat des Saarlandes, Saarbrucken, 1997.
27. V. Miller, Use of elliptic curves in cryptography, Advances in Cryptology - CRYPTO '85, Lecture Notes in Computer Science, vol. 218, Springer-Verlag, Berlin, 1986, pp. 417126.
28. S. Paulus, An algorithm of subexponential type computing the class group of quadratic orders over principal ideal domains, Algorithmic Number Theory - ANTSII (Université Bordeaux I, Talence), Lecture Notes in Computer Science, vol. 1122, Springer-Verlag, Berlin, 1996, pp. 243-257.
29. R. Scheidler, J. Buchmann, and H.C. Williams, A key-exchange protocol using real quadratic fields, J. Cryptology 7 (1994), 171-199.
30. M. Seysen, A probabilistic factorization algorithm with quadratic forms of negative discriminant, Math. Comp. 48 (1987), 757-780.
31. D. Shanks, The infrastructure of real quadratic fields and its applications, Proc. 1972 Number Theory Conference, Boulder, Colorado, 1972, pp. 217-224.
32. R.D. Silverman, The multiple polynomial quadratic sieve, Math. Comp. 48 (1987), 329-339.