Multivariates Polynomials for Hashing

  • Jintai Ding
  • Bo-Yin Yang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4990)

Abstract

We propose the idea of building a secure hash using quadratic or higher degree multivariate polynomials over a finite field as the compression function. We analyze some security properties and potential feasibility, where the compression functions are randomly chosen high-degree polynomials, and show that under some plausible assumptions, high-degree polynomials as compression functions has good properties. Next, we propose to improve on the efficiency of the system by using some specially designed polynomials generated by a small number of random parameters, where the security of the system would then relies on stronger assumptions, and we give empirical evidence for the validity of using such polynomials.

Keywords

hash function multivariate polynomials sparse 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aumasson, J.-P., Meier, W.: Analysis of multivariate hash functions. In: Proc. ICISC. LNCS (to appear, 2007), cf.http://www.131002.net/files/pub/AM07.pdf
  2. 2.
    Bardet, M., Faugère, J.-C., Salvy, B.: On the complexity of Gröbner basis computation of semi-regular overdetermined algebraic equations. In: Proceedings of the International Conference on Polynomial System Solving, pp. 71–74 (2004); Previously INRIA report RR-5049 Google Scholar
  3. 3.
    Berbain, C., Gilbert, H., Patarin, J.: QUAD: A Practical Stream Cipher with Provable Security. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 109–128. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Computational Algebra Group, University of Sydney. The MAGMA Computational Algebra System for Algebra, Number Theory and Geometry http://magma.maths.usyd.edu.au/magma/
  5. 5.
    Courtois, N., Goubin, L., Meier, W., Tacier, J.-D.: Solving Underdefined Systems of Multivariate Quadratic Equations. In: Naccache, D., Paillier, P. (eds.) PKC 2002. LNCS, vol. 2274, Springer, Heidelberg (2002)CrossRefGoogle Scholar
  6. 6.
    Ding, J., Gower, J., Schmidt, D.: Multivariate Public-Key Cryptosystems. In: Advances in Information Security, Springer, Heidelberg (2006)Google Scholar
  7. 7.
    Garey, M.R., Johnson, D.S.: Computers and Intractability — A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979)MATHGoogle Scholar
  8. 8.
    Kipnis, A., Shamir, A.: Cryptanalysis of the HFE Public Key Cryptosystem by Relinearization. In: Wiener, M.J. (ed.) CRYPTO 1999. LNCS, vol. 1666, Springer, Heidelberg (1999), http://www.minrank.org/hfesubreg.ps or http://citeseer.nj.nec.com/kipnis99cryptanalysis.html Google Scholar
  9. 9.
    Menezes, A., Koblitz, N.: Another Look at “Provable Security”. II. In: Barua, R., Lange, T. (eds.) INDOCRYPT 2006. LNCS, vol. 4329, pp. 148–175. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Koblitz, N., Menezes, A.: Another look at “provable security”. Journal of Cryptology 20, 3–37 (2004)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Raddum, H., Semaev, I.: New technique for solving sparse equation systems. Cryptology ePrint Archive, Report 2006/475 (2006), http://eprint.iacr.org/
  12. 12.
    Sugita, M., Kawazoe, M., Imai, H.: Gröbner basis based cryptanalysis of sha-1. Cryptology ePrint Archive, Report 2006/098(2006), http://eprint.iacr.org/
  13. 13.
    Wang, X., Yin, Y.L., Yu, H.: Finding Collisions in the Full SHA-1. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 17–36. Springer, Heidelberg (2005)Google Scholar
  14. 14.
    Wang, X., Yu, H.: How to break md5 and other hash functions. In: Cramer, R.J.F. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 19–35. Springer, Heidelberg (2005)Google Scholar
  15. 15.
    Wolf, C., Preneel, B.: Taxonomy of public key schemes based on the problem of multivariate quadratic equations. Cryptology ePrint Archive, Report 2005/077, 64 (May, 2005) http://eprint.iacr.org/2005/077/
  16. 16.
    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)Google Scholar
  17. 17.
    Yang, B.-Y., Chen, O.C.-H., Bernstein, D.J., Chen, J.-M.: Analysis of QUAD. In: Biryukov, A. (ed.) Fast Software Encryption — FSE 2007. volume to appear of Lecture Notes in Computer Science, pp. 302–319. Springer, Heidelberg (2007) workshop record available Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Jintai Ding
    • 1
  • Bo-Yin Yang
    • 2
  1. 1.University of Cincinnati and Technische Universität Darmstadt 
  2. 2.Institute of Information Science, Academia Sinica 

Personalised recommendations