Efficient generation of secure elliptic curves

International Journal of Information Security

Volumn 6, Issue 1, 2007, Pages 47-63

  Konstantinou, Elisavet ^a,b   Stamatiou, Yannis C ^a,c   Zaroliagis, Christos ^a,d  

^a University of Patras and Computer Technology Institute   (Greece)

^b UNIVERSITY OF THE AEGEAN   (Greece)

^c UNIVERSITY OF IOANNINA   (Greece)

^d UNIVERSITY OF PATRAS   (Greece)

Author keywords

Complex Multiplication; Elliptic curve cryptosystems; Public key cryptography; Weber polynomials

Indexed keywords

EID: 33845707856     PISSN: 16155262     EISSN: 16155270     Source Type: Journal    
DOI: 10.1007/s10207-006-0009-3     Document Type: Article

References (38)

1. Argyroudis P.: NTRG ECC-LIB WINCE'a WinCE port of ECC-LIB, available at: http://www.ntrg.cs.tcd.ie/~argp/software/ntrg-ecc-lib-wince.html (2004)
2. Atkin A.O.L. and Morain F. (1993). Elliptic curves and primality proving. Math. Comput. 61: 29-67
3. Baier, H. (2002). Efficient algorithms for generating elliptic curves over finite fields suitable for use in cryptography. PhD Thesis, Deptartment of Computer Science, Technical University of Darmstadt
4. Baier, H., Buchmann, J.: Efficient construction of cryptographically strong elliptic curves. In: Progress in Cryptology - INDOCRYPT 2000, LNCS, vol. 1977, pp. 191-202. Springer, Berlin Heidelberg New York (2000)
5. Berlekamp E.R. (1970). Factoring polynomials over large finite fields. Math Comput 24: 713-735
6. Blake I., Seroussi G. and Smart N. (1999). Elliptic curves in cryptography London Mathematical Society, Lecture Note Series 265. Cambridge University Press, Cambridge
7. Burton, D.: Elementary Number Theory, 4th edn. McGraw-Hill, New York (1998)
8. h = P. Giornale di Matematiche di Battaglini 46, 33-90 (1908)
9. Enge, A., Morain, F.: Comparing invariants for class fields of imaginary quadratic fields. In: Algorithmic Number Theory'-ANTS-V. Lecture Notes in Computer Science, vol. 2369, pp. 252-266. Springer, Berlin Heidelberg New York (2002)
10. Enge, A., Schertz, R.: Constructing elliptic curves from modular curves of positive genus. (preprint 2003)
11. Frey G. and Rück H.G. (1994). A remark concerning m -divisibility and the discrete logarithm problem in the divisor class group of curves. Math Comput 62: 865-874
12. GNU multiple precision library, 3.1.1. edn. Available at: http://www.swox.com/gmp (2000)
13. Gura, N., Eberle, H., Shantz, S.C.: Generic implementations of elliptic curve cryptography using partial reduction. In: Proceedings of the 9th ACM Conference on Computer and Communications Security-CCS'02, pp. 108-116
14. Hankerson D., Menezes A. and Vanstone S. (2004). Guide to elliptic curve cryptography. Springer, Berlin Heidelberg New York
15. Herzberg, A., Jakobsson, M., Jarecki, S., Krawczyk, H., Yung, M.: Proactive public key and signature systems. In: Proceedings of the 4th ACM Conference on Computer and Communications Security-CCS'97, pp. 100-110
16. IEEE P1363/D13: Standard specifications for public-key cryptography, ballot draft. http://www.grouper.ieee.org/groups/1363/tradPK/draft.html (1999)
17. Kaltofen, E., Yui, N.: Explicit construction of the Hilbert class fields of imaginary quadratic fields by integer lattice reduction. Research Report 89-13, Renseelaer Polytechnic Institute (1989)
18. Kaltofen, E., Valente, T., Yui, N.: An improved Las Vegas primality test. In: Proceedings of the ACM-SIGSAM 1989 International Symposium on Symbolic and Algebraic Computation, pp. 26-33 (1989)
19. Konstantinou, E., Stamatiou, Y., Zaroliagis, C.: A software library for elliptic curve cryptography. In: Proceedings of the 10th European Symposium on Algorithms-ESA 2002 (Engineering and Applications Track). Lecture Notes in Computer Science, vol. 2461, pp. 625-637. Springer, Berlin Heidelberg New York (2002)
20. Konstantinou, E., Stamatiou, Y., Zaroliagis, C.: On the efficient generation of elliptic curves over prime fields. In: Cryptographic hardware and embedded systems-CHES 2002. Lecture Notes in Computer Science, vol. 2523, pp. 333-348. Springer, Berlin Heidelberg New York (2002)
21. Konstantinou, E., Stamatiou, Y., Zaroliagis, C.: On the Use of Weber polynomials in elliptic curve cryptography. In: Public key infrastructure-EuroPKI 2004. Lecture Notes in Computer Science, vol. 3093, pp. 335-349. Springer, Berlin Heidelberg New York, (2003)
22. Lay, G.J., Zimmer, H.: Constructing elliptic curves with given group order over large finite fields. In: Algorithmic number theory-ANTS-I. Lecture Notes in Computer Science, vol. 877, pp. 250-263. Springer, Berlin Heidelberg New York (1994)
23. LiDIA: A library for computational number theory, Technical University of Darmstadt. Available from http://www.informatik.tu-darmstadt.de/TI/ LiDIA/Welcome.html (2001)
24. Menezes A.J., Okamoto T. and Vanstone S.A. (1993). Reducing elliptic curve logarithms to a finite field. IEEE Trans. Info. Theory 39: 1639-1646
25. Müller, V., Paulus, S.: On the generation of cryptographically strong elliptic curves (preprint 1997)
26. Niven, I., Zuckerman, H.S., Montgomery, H.L.: An introduction to the theory of numbers, 5th edn. Wiley, New York (1991)
27. Pohlig G.C. and Hellman M.E. (1978). An improved algorithm for computing logarithms over gf (p) and its cryptographic significance. IEEE Trans. Info. Theory 24: 106-110
28. Satoh T. and Araki K. (1998). Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves. Comm. Math. Univ. Sancti Pauli 47: 81-91
29. Savas, E., Schmidt, T.A., Koc, C.K.: Generating elliptic curves of prime order. In: Cryptographic hardware and embedded systems-CHES 2001. Lecture Notes in Computer Science, vol. 2162, pp. 145-161. Springer, Berlin Heidelberg New York (2001)
30. Schertz R. (2002). Weber's class invariants revisited. J. Thor. Nombres Bordeaux 14: 1
31. Semaev I.A. (1998). Evaluation of discrete logarithms on some elliptic curves. Math. Comput. 67: 353-356
32. Silverman J.H. (1986). The arithmetic of elliptic curves, GTM 106. Springer, Berlin Heidelberg New York
33. Smart N.P. (1999). The discrete logarithm problem on elliptic curves of trace one. J. Cryptogr. 12: 193-196
34. Spallek, A.-M.: Konstruktion einer elliptischen Kurve über einem endlichen Körper zu gegebener Punktegruppe. Master Thesis, Universität GH Essen (1992)
35. Valente, T.: A distributed approach to proving large numbers prime. Rensselaer Polytechnic Institute Troy, New York, Thesis (1992)
36. Washington L.C. (2003). Elliptic curves: Number theory and cryptography. Chapman & Hall/CRC, Boca Raton
37. Weber, H.: Algebra III. Vieweg (1908)
38. WilliamsH P.: Available at: http://www.mindspring.com/~pate