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Systemsicherheit pp 370–383Cite as

A Survey of Cryptosystems Based on Imaginary Quadratic Orders

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Part of the book series: DuD-Fachbeiträge ((DUD))

Abstract

Since nobody can guarantee that popular public key cryptosystems based on factoring or the computation of discrete logarithms in some group will stay secure forever, it is important to study different primitives and groups which may be utilized if a popular class of cryptosystems gets broken.

A promising candidate for a group in which the DL-problem seems to be hard is the class group’ Cl(Δ) of an imaginary quadratic order, as proposed by Buchmann and Williams [BuWi88].Recently this type of group has obtained much attention, because there was proposed a very efficient cryptosystem based on non-maximal imaginary quadratic orders [PaTa98a], later on called NICE (for New Ideal Coset Encryption) with quadratic decryption time. To our knowledge this is the only scheme having this property. First implementations show that the time for decryption is comparable to RS A encryption with e = 216 +1. Very recently there was proposed an efficient NICE-Schnorr type signature scheme [HuMe99] for which the signature generation is more than twice as fast as in the original scheme based on F*p.

Due to these results there has been increasing interest in cryptosystems based on imaginary quadratic orders. Therefore it seems necessary to provide an up to date survey to facilitate further work in this direction. Our survey will discuss the history, the state of the art and future directions of cryptosystems based on imaginary quadratic orders.

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References

  1. I. Biehl, J. Buchmann: An analysis of the reduction algorithms for binary quadratic forms, in P. Engel, H. Syta (Ed.): Voronoi’s Impact on Modern Science, Vol. 1, Institute of Mathematics of National Academy of Sciences, Kyiv, Ukraine, 1998.

    Google Scholar 

  2. I. Biehl, J. Buchmann, S. Hamdy, A. Meyer: Cryptographic Protocols Based on the Intractibility of Extracting Roots and Computing Discrete Logarithms, Technical Report, University of Technology, Darmstadt, 1999. http://www.informatik.tu-darmstadt.de/TI/Veroeffentlichung/TR/ Welcome.html"

    Google Scholar 

  3. I. Biehl, S. Paulus, T. Takagi: An efficient undeniable signature scheme based on non-maximal imaginary quadratic orders, Technical Report, University of Technology, Darmstadt, 1999. http://www.informatik.tu-darmstadt.de/TI/ Veroeffentlichung/TR/Welcome. html

    Google Scholar 

  4. Z.I. Borevich, I.R. Shafarevich: Number Theory Academic Press: New York, 1966.

    Google Scholar 

  5. R. Brent: ECM champs. ftp://ftp.comlab.ox.ac.uk/pub/Documents/ techpapers/Richard.Brent/champs.ecm

    Google Scholar 

  6. E. Brickell, D. Gordon, K. McCurley, D. Wilson: Fast Exponentiation with Precomputation, Proceedings of Eurocrypt’ 92, Springer LNCS 658, 1993, S. 200–207.

    Google Scholar 

  7. J. Buchmann, S. Düllmann: On the computation of discrete logarithms in class groups, Advances in Cryptology — CRYPTO’ 90, Springer LNCS 773, 1991, S. 134–139.

    Google Scholar 

  8. J. Buchmann, S. Düllmann, H.C. Williams: On the complexity and efficiency of a new key exchange system, Advances in Cryptology — EUROCRYPT’ 89, Springer LNCS 434, 1990, S. 597–616.

    Google Scholar 

  9. J. Buchmann, H.C. Williams: A key-exchange system based on imagninary quadratic fields. Journal of Cryptology Vol. 1, 1988, S. 107–118.

    Article  MathSciNet  MATH  Google Scholar 

  10. D.A. Buell: Binary Quadratic Forms — Classical Theory and Modern Computations, Springer, 1989.

    Google Scholar 

  11. J. Cowie, B. Dodson, M. Elkenbracht-Huizing, A.K. Lenstra, P.L. Montgomery, J. Zayer: A worldwide number field sieve factoring record: on to 512 bits, proceedings of ASIACRYPT’96, Springer LNCS 1163, 1996, S. 382–394.

    MathSciNet  Google Scholar 

  12. H. Cohen: A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics 138, Springer, 1993.

    Google Scholar 

  13. D. Coppersmith, A.M. Odlyzko, R. Schroeppel: Discrete logarithms in GF(p), Algorithmica, Vol. 1, 1986, S. 1–15.

    Article  MathSciNet  MATH  Google Scholar 

  14. D.A. Cox: Primes of the form x 2 + ny 2, John Wiley & Sons, 1989.

    Google Scholar 

  15. W. Diffie, M. Hellman: New directions in cryptography, IEEE Transactions on Information Theory Vol. 22, 1976, S. 472–492.

    Article  MathSciNet  Google Scholar 

  16. S. Düllmann: Ein neues Verfahren zum öffentlichen Schlüsselaustausch, Diplomarbeit, Universit”at Düsseldorf, 1988.

    Google Scholar 

  17. S. Düllmann: Ein Algorithmus zur Bestimmung der Klassenzahl positiv def-initer binärer quadratischer Formen, Dissertation, Universit”at Saarbrücken, 1991.

    Google Scholar 

  18. A. Fiat, A. Shamir: How to prove yourself: Practical solutions to identification and signature problems, Advances in Cryptology, Proceedings of CRYPTO’ 86, Springer LNCS 263, 1987, S. 186–194.

    MathSciNet  Google Scholar 

  19. C.F. Gau”s: Disquisitiones Arithmeticae, 1801, reprinted 1986 by Springer, ISBN 0-387-96254-9.

    Google Scholar 

  20. D.M. Gordon: Discrete logarithms in GF(p) using the number field sieve, SIAM Journal on Discrete Mathematics Vol. 6, 1993, S. 124–138.

    Article  MathSciNet  MATH  Google Scholar 

  21. S. Hamdy: The key-length of DL-based cryptosystems in class groups, 1999.

    Google Scholar 

  22. J.L. Hafner, K.S. McCurley: A rigorous subexponential algorithm for computation of class groups, Journal of the American Mathematical Society, Vol. 2, 1989, S. 837–850.

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Hartmann, S. Paulus, T. Takagi: NICE — New Ideal Coset Encryption, CHES, erscheint in Springer LNCS, 1999. www.informatik.tu-darmstadt.de/TI/Veroeffentlichung/TR/Welcome.html

    Google Scholar 

  24. L.K. Hua: Introduction to Number Theory. Springer, 1982.

    Google Scholar 

  25. D. Hühnlein, M.J. Jacobson, S. Paulus, T. Takagi: A cryptosystem based on non-maximal imaginary quadratic orders with fast decryption, Advances in Cryptology — EUROCRYPT’ 98, Springer LNCS 1403, 1998, S. 294–307.

    Article  Google Scholar 

  26. D. Hühnlein, A. Meyer, T. Takagi: Rabin and RSA analogues based on non-maximal imaginary quadratic orders, Proceedings of ICICS’ 98, 1998, S. 221–240.

    Google Scholar 

  27. D. Hühnlein: Efficient implementation of cryptosystems based on non-maximal imaginary quadratic orders, erscheint in Proceedings of SAC’99, Springer LNCS 1758, 2000, S. 150–167, www.informatik.tu-darmstadt.de/TI/Veroeffentlichung/TR/Welcome.html"

    Google Scholar 

  28. D. Hühnlein, J. Merkle: An efficient NICE-Schnorr-type cryptosystem, erscheint in PKC2000, Melbourne, Januar 2000, Springer LNCS. http://www. informatik.tu-darmstadt.de/TI/Veroeffentlichung/TR/Welcome.html

    Google Scholar 

  29. D. Hühnlein, T. Takagi: Reducing logarithms in totally non-maximal imaginary quadratic orders to logarithms in finite fields, Advances in Cryptology — Asiacrypt’99, Springer LNCS 1716, 1999, S. 219.

    Article  Google Scholar 

  30. M.J. Jacobson Jr.: Subexponential Class Group Computation in Quadratic Orders, Berichte aus der Informatik, Shaker, ISBN 3-8265-6374-3, 1999.

    Google Scholar 

  31. M. Joye, J.J. Quisquater: On Rabin-type signatures, Research contribution to IEEE-P1363, 1999. http://grouper.ieee.org/groups/1363/contrib.html

    Google Scholar 

  32. H.W. Lenstra: On the computation of regulators and class numbers of quadratic fields, London Math. Soc. Lecture Notes, Vol. 56, 1982, S. 123–150.

    MathSciNet  Google Scholar 

  33. H.W. Lenstra: Factoring integers with elliptic curves, Annals of Mathematics, Vol. 126, 1987, S. 649–673.

    Article  MathSciNet  MATH  Google Scholar 

  34. A.K. Lenstra, H.W. Lenstra Jr. (Ed.): The development of the number field sieve, Lecture Notes in Mathematics, Springer, 1993.

    Google Scholar 

  35. H.W. Lenstra: Complex Multiplication Structure of Elliptic Curves, Journal of Number Theory, Vol. 56, No. 2, 1996, S. 227–241.

    Article  MathSciNet  MATH  Google Scholar 

  36. LiDIA: A c++ library for algorithmic number theory, http://www. informatik.tu-darmstadt.de/TI/LiDIA

    Google Scholar 

  37. U. Maurer, Y. Yacobi: A non-interactive public-key distribution system, Design Codes and Cryptography, No. 9, 1996, S. 305–316.

    MathSciNet  MATH  Google Scholar 

  38. K.S. McCurley: Cryptographic key distribution and computation in class groups, Number Theory and applications, NATO ASI series, Series C, Vol. 265, Dordrecht, 1989, S. 459–479.

    MathSciNet  Google Scholar 

  39. A. Meyer: Ein neues Identifikations-und Signaturverfahren über imaginärquadratischen Zahlkörpern, Diplomarbeit, Universit”at Saarbrücken, 1997. ftp://ftp.informatik.tu-darmstadt.de/pub/TI/reports/amy.diplom.ps.gz

    Google Scholar 

  40. National Institute of Standards and Technology (NIST): Digital Signature Standard (DSS). Federal Information Processing Standards Publication 186, FIPS-186, 19. Mai 1994.

    Google Scholar 

  41. J. Neukirch, Algebraische Zahlentheorie, Springer, 1992.

    Google Scholar 

  42. S. Paulus, T. Takagi: A new public key cryptosystem with quadratic decryption time, erscheint in Journal of Cryptology, 1998. http://www.informatik. tu-darmstadt.de/TI/Mitarbeiter/sachar.html

    Google Scholar 

  43. S. Paulus, T. Takagi: A generalization of the Diffie-Hellman problem based on the coset problem allowing fast decryption, Proceedings of ICICS’ 98, 1998.

    Google Scholar 

  44. R. Peralta, E. Okamoto: Faster factoring of integers of a special form, IEICE Trans. Fundamentals, Vol. E-79-A, No. 4, 1996, S. 489–493.

    Google Scholar 

  45. S. Cavallar, B. Dodson, A. Lenstra, P. Leyland, W. Lioen, P.L. Montgomery, B. Murphy, H. te Riele, P. Zimmerman: Factorization of RSA-140 Using the Number Field Sieve, Proceedings of ASIACRYPT’99, Springer LNCS 1716, 1999, S. 195–207.

    Google Scholar 

  46. H. te Riele & al.: Factorization of RSA-155 with the Number Field Sieve, posting in sci.crypt.research, August 1999.

    Google Scholar 

  47. R. Rivest, A. Shamir, L. Adleman: A method for obtaining digital signatures and public key-cryptosystems, Communications of the ACM, Vol. 21, 1978, S. 120–126.

    Article  MathSciNet  MATH  Google Scholar 

  48. M. Seysen: A probabilistic factoring algorithm with quadratic forms of negative discriminant, Math. Comp. 48, 1987, S. 737–780.

    Article  MathSciNet  Google Scholar 

  49. R.D. Silverman: The multiple polynomial quadratic sieve, Math. Comp. 48, 1987, S. 329–229.

    Article  MathSciNet  MATH  Google Scholar 

  50. R.J. Schoof: Quadratic Fields and Factorization. In: H.W. Lenstra, R. Ti-jdeman (Ed.): Computational Methods in Number Theory. Math. Centrum Tracts 155, Part II, Amsterdam, 1983, S. 235–286.

    Google Scholar 

  51. C.P. Schnorr, H.W. Lenstra: A Monte Carlo factoring algorithm with linear storage, Mathematics of Computation, Vol. 43, 1984, S. 289–312.

    Article  MathSciNet  MATH  Google Scholar 

  52. D. Shanks: Gauss’ ternary form reduction and the 2-Sylow subgroup, Math. Comp. 25, 1971, S. 837–853.

    MathSciNet  MATH  Google Scholar 

  53. D. Shanks: Class number, a theory of factorization and genera, Proc. Symposium Pure Mathematics, American Mathematical Society 20, 1971, S. 415–440.

    MathSciNet  Google Scholar 

  54. D. Weber: Computing discrete logarithms with quadratic number rings, Advances in Cryptology — EUROCRYPT’ 98, Springer LNCS 1403, 1998, S. 171–183.

    Article  Google Scholar 

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  1. Secunet Security Networks AG, Eschborn, Deutschland

    Detlef Hühnlein

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Patrick Horster

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© 2000 Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden

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Hühnlein, D. (2000). A Survey of Cryptosystems Based on Imaginary Quadratic Orders. In: Horster, P. (eds) Systemsicherheit. DuD-Fachbeiträge. Vieweg+Teubner Verlag. https://doi.org/10.1007/978-3-322-84957-1_30

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  • DOI: https://doi.org/10.1007/978-3-322-84957-1_30

  • Publisher Name: Vieweg+Teubner Verlag

  • Print ISBN: 978-3-322-84958-8

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