Advertisement

Building Secure Tame-like Multivariate Public-Key Cryptosystems: The New TTS

  • Bo-Yin Yang
  • Jiun-Ming Chen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3574)

Abstract

Multivariate public-key cryptosystems (sometimes polynomial-based PKC’s or just multivariates) handle polynomials of many variables over relatively small fields instead of elements of a large ring or group. The “tame-like” or “sparse” class of multivariates are distinguished by the relatively few terms that they have per central equation. We explain how they differ from the “big-field” type of multivariates, represented by derivatives of C  ∗  and HFE, how they are better, and give basic security criteria for them. The last is shown to be satisfied by efficient schemes called “Enhanced TTS” which is built on a combination of the Oil-and-Vinegar and Triangular ideas. Their security levels are estimated. In this process we summarize and in some cases, improve rank-based attacks, which seek linear combinations of certain matrices at given ranks. These attacks are responsible for breaking many prior multivariate designs.

Keywords

Smart Card Signature Scheme Central Equation Digital Signature Scheme Multivariate Signature Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bardet, M., Faugère, J.-C., Salvy, B., Yang, B.-Y.: Asymptotic Expansion of the Degree of Regularity for Semi-Regular Systems of Equations. In: To be presented MEGA 2005 (2005)Google Scholar
  2. 2.
    Braeken, A., Wolf, C., Preneel, B.: A Study of the Security of Unbalanced Oil and Vinegar Signature Schemes. In: Menezes, A. (ed.) CT-RSA 2005. LNCS, vol. 3376, pp. 29–43. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  3. 3.
    Chen, J.-M., Yang, B.-Y.: A More Secure and Efficacious TTS Scheme. In: Lim, J.-I., Lee, D.-H. (eds.) ICISC 2003. LNCS, vol. 2971, pp. 320–338. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  4. 4.
    Chou, C.-Y., Hu, Y.-H., Lai, F.-P., Wang, L.-C., Yang, B.-Y.: Tractable Rational Map Signature. In: Vaudenay, S. (ed.) PKC 2005. LNCS, vol. 3386, pp. 244–257. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  5. 5.
    Coppersmith, D., Stern, J., Vaudenay, S.: Attacks on the Birational Permutation Signature Schemes. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 435–443. Springer, Heidelberg (1994)Google Scholar
  6. 6.
    Courtois, N., Goubin, L., Meier, W., Tacier, J.: Solving Underdefined Systems of Multivariate Quadratic Equations. In: Naccache, D., Paillier, P. (eds.) PKC 2002. LNCS, vol. 2274, pp. 211–227. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. 7.
    Courtois, N., Klimov, A., Patarin, J., Shamir, A.: Efficient Algorithms for Solving Overdefined Systems of Multivariate Polynomial Equations. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 392–407. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  8. 8.
    Diem, C.: The XL-algorithm and a conjecture from commutative algebra. In: Lee, P.J. (ed.) ASIACRYPT 2004. LNCS, vol. 3329, pp. 338–353. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  9. 9.
    Ding, J., Schmidt, D.: Rainbow, a new multivariate polynomial signature system. In: Ioannidis, J., Keromytis, A.D., Yung, M. (eds.) ACNS 2005. LNCS, vol. 3531, pp. 164–175. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Ding, J., Schmidt, D.: Cryptanalysis of HFEv and Internal Perturbation of HFE. In: Vaudenay, S. (ed.) PKC 2005. LNCS, vol. 3386, pp. 288–301. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Ding, J., Yin, Z.: Cryptanalysis of TTS and Tame-like Multivariable Signature Schemes. In: Presentation at IWAP 2004 (2004)Google Scholar
  12. 12.
    Faugère, J.-C.: A New Efficient Algorithm for Computing Gröbner Bases without Reduction to Zero (F5). In: Proceedings of ISSAC, ACM Press, New York (2002)Google Scholar
  13. 13.
    Faugère, J.-C., Joux, A.: Algebraic Cryptanalysis of Hidden Field Equations (HFE) Cryptosystems Using Gröbner Bases. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 44–60. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  14. 14.
    Feldhofer, M., Dominikus, S., Wolkerstorfer, J.: Strong Authentication for RFID Systems Using the AES Algorithm. In: Joye, M., Quisquater, J.-J. (eds.) CHES 2004. LNCS, vol. 3156, pp. 357–370. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  15. 15.
    Garey, M., Johnson, D.: Computers and Intractability, A Guide to the Theory of NP-completeness, p. 251. Freeman and Co., New York (1979)zbMATHGoogle Scholar
  16. 16.
    Goubin, L., Courtois, N.: Cryptanalysis of the TTM Cryptosystem. In: Okamoto, T. (ed.) ASIACRYPT 2000. LNCS, vol. 1976, pp. 44–57. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  17. 17.
    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proc. 28th Annual ACM Symposium on the Theory of Computing, May 1996, pp. 212–220 (1996)Google Scholar
  18. 18.
    Joux, A., Kunz-Jacques, S., Muller, F., Ricordel, P.-M.: Cryptanalysis of the Tractable Rational Map Cryptosystem. In: Vaudenay, S. (ed.) PKC 2005. LNCS, vol. 3386, pp. 258–274. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  19. 19.
    Kipnis, A., Patarin, J., Goubin, L.: Unbalanced Oil and Vinegar Signature Schemes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 206–222. Springer, Heidelberg (1999)Google Scholar
  20. 20.
    Kipnis, A., Shamir, A.: Cryptanalysis of the Oil and Vinegar Signature Scheme. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 257–266. Springer, Heidelberg (1998)Google Scholar
  21. 21.
    Kipnis, A., Shamir, A.: Cryptanalysis of the HFE Public Key Cryptosystem. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 19–30. Springer, Heidelberg (1999)Google Scholar
  22. 22.
    Matsumoto, T., Imai, H.: Public Quadratic Polynomial-Tuples for Efficient Signature-Verification and Message-Encryption. In: Günther, C.G. (ed.) EUROCRYPT 1988. LNCS, vol. 330, pp. 419–453. Springer, Heidelberg (1988)Google Scholar
  23. 23.
    Moh, T.: A Public Key System with Signature and Master Key Functions. Communications in Algebra 27, 2207–2222 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
  25. 25.
    Patarin, J.: Cryptanalysis of the Matsumoto and Imai Public Key Scheme of Eurocrypt 1988. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 248–261. Springer, Heidelberg (1995)Google Scholar
  26. 26.
    Patarin, J.: Hidden Fields Equations (HFE) and Isomorphisms of Polynomials (IP): Two New Families of Asymmetric Algorithms. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 33–48. Springer, Heidelberg (1996)Google Scholar
  27. 27.
    Patarin, J., Courtois, N., Goubin, L.: FLASH, a Fast Multivariate Signature Algorithm. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 298–307. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  28. 28.
    Shor, P.W.: Algorithms for quantum computation: Discrete logarithms and factoring. In: Proc. 35th Ann. Symp. on Foundations of Comp. Sci., pp. 124–134. IEEE Comp. Soc. Press, Los Alamitos (1994)CrossRefGoogle Scholar
  29. 29.
    Wang, L.: Tractable Rational Map Cryptosystem, see ePrint 2004/046Google Scholar
  30. 30.
    Weisstein, E.: RSA-576 Factored, mathworld.wolfram.com/news/2003-12-05/rsa
  31. 31.
    Wolf, C.: Efficient Public Key Generation for Multivariate Cryptosystems. In: Int’l Workshop on Cryptographic Algorithms and their Uses 2004, pp. 78–93 (2004), also ePrint 2003/089Google Scholar
  32. 32.
    Wolf, C., Braeken, A., Preneel, B.: Efficient Cryptanalysis of RSE(2)PKC and RSSE(2)PKC. In: Blundo, C., Cimato, S. (eds.) SCN 2004. LNCS, vol. 3352, pp. 294–309. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  33. 33.
    Wolf, C., Preneel, B.: Taxonomy of Public-Key Schemes based on the Problem of Multivariate Quadratic Equations, ePrint 2005/077Google Scholar
  34. 34.
    Yang, B.-Y., Chen, J.-M.: All in the XL Family: Theory and Practice. In: Park, C.-s., Chee, S. (eds.) ICISC 2004. LNCS, vol. 3506, pp. 67–86. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  35. 35.
    Yang, B.-Y., Chen, J.-M., Chen, Y.-H.: TTS: High-Speed Signatures from Low- End Smartcards. In: Joye, M., Quisquater, J.-J. (eds.) CHES 2004. LNCS, vol. 3156, pp. 371–385. Springer, Heidelberg (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Bo-Yin Yang
    • 1
  • Jiun-Ming Chen
    • 2
  1. 1.Dept. of MathematicsTamkang UniversityTamsuiTaiwan
  2. 2.Chinese Data Security, Inc., & Nat’l Taiwan U.Taipei

Personalised recommendations