Inverting HFE Systems Is Quasi-Polynomial for All Fields

  • Jintai Ding
  • Timothy J. Hodges
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6841)


In this paper, we present and prove the first closed formula bounding the degree of regularity of an HFE system over an arbitrary finite field. Though these bounds are not necessarily optimal, they can be used to deduce

  1. 1

    if D, the degree of the corresponding HFE polynomial, and q, the size of the corresponding finite field, are fixed, inverting HFE system is polynomial for all fields;

  2. 2

    if D is of the scale O(n α ) where n is the number of variables in an HFE system, and q is fixed, inverting HFE systems is quasi-polynomial for all fields.


We generalize and prove rigorously similar results by Granboulan, Joux and Stern in the case when q = 2 that were communicated at Crypto 2006.


Quantum Computer Quotient Ring Quadratic Element Quadratic Operator Galois Ring 
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.


  1. 1.
    Bardet, M., Faugère, J.-C., Salvy, B.: On the complexity of Gröbner basis computation of semi-regular overdetermined algebraic equations. In: International Conference on Polynomial System Solving - ICPSS, pp. 71–75 (November 2004)Google Scholar
  2. 2.
    Bettale, L., Faugère, J.-C., Perret, L.: Cryptanalysis of Multivariate and Odd-Characteristic HFE Variants. In: Catalano, D., Fazio, N., Gennaro, R., Nicolosi, A. (eds.) PKC 2011. LNCS, vol. 6571, pp. 441–458. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    Ding, J.: Mutants and its impact on polynomial solving strategies and algorithms. Privately distributed research note, University of Cincinnati and Technical University of Darmstadt (2006)Google Scholar
  4. 4.
    Ding, J., Buchmann, J., Mohamed, M., Mohamed, W., Weinmann, R.-P.: Mutant XL. In: First International Conference on Symbolic Computation and Cryptography – SCC (2008)Google Scholar
  5. 5.
    Ding, J., Gower, J., Schmidt, D.: Multivariate Public Key Cryptography. Advances in Information Security series. Springer, Heidelberg (2006)Google Scholar
  6. 6.
    Ding, J., Hodges, T.J., Kruglov, V.: Growth of the ideal generated by a quadratic boolean function. In: Sendrier, N. (ed.) PQCrypto 2010. LNCS, vol. 6061, pp. 13–27. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  7. 7.
    Ding, J., Hodges, T.J., Kruglov, V., Schmidt, D., Tohaneanu, S.: Growth of the ideal generated by a multivariate quadratic function over GF(3), preprintGoogle Scholar
  8. 8.
    Ding, J., Schmidt, D., Werner, F.: Algebraic attack on HFE revisited. In: Wu, T.-C., Lei, C.-L., Rijmen, V., Lee, D.-T. (eds.) ISC 2008. LNCS, vol. 5222, pp. 215–227. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  9. 9.
    Dubois, V., Gama, N.: The degree of regularity of HFE systems. In: Abe, M. (ed.) ASIACRYPT 2010. LNCS, vol. 6477, pp. 557–576. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Faugère, J.-C., Joux, A.: Algebraic cryptanalysis of hidden field equation (HFE) cryptosystems using gröbner bases. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 44–60. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  11. 11.
    Garey, M.R., Johnson, D.S.: Computers and intractability, A Guide to the theory of NP-completeness. W.H. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  12. 12.
    Granboulan, L., Joux, A., Stern, J.: Inverting HFE Is Quasipolynomial. In: Dwork, C. (ed.) CRYPTO 2006. LNCS, vol. 4117, pp. 345–356. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Kipnis, A., Shamir, A.: Cryptanalysis of the HFE Public Key Cryptosystem by Relinearization. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 19–30. Springer, Heidelberg (1999)Google Scholar
  14. 14.
    Lidl, R., Niederreiter, H.: Finite Fields, Encyclopedia of Mathematics and its Applications, vol. 20. Cambridge University Press, Cambridge (1997)Google Scholar
  15. 15.
    Mohamed, M.S.E., Cabarcas, D., Ding, J., Buchmann, J., Bulygin, S.: MXL3: An Efficient Algorithm for Computing Gröbner Bases of Zero-Dimensional Ideals. In: Lee, D., Hong, S. (eds.) ICISC 2009. LNCS, vol. 5984, pp. 87–100. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  16. 16.
    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
  17. 17.
    O’Meara, O.T.: Introduction to Quadratic Forms. Springer, Berlin (1963)zbMATHGoogle Scholar
  18. 18.
    Patarin, J.: Cryptanalysis of the Matsumoto and Imai Public Key Scheme of Eurocrypt ’88. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 248–261. Springer, Heidelberg (1995)Google Scholar
  19. 19.
    Shor, P.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev. 41(2), 303–332 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Wan, Z.-X.: Lectures on Finite Fields and Galois Rings. World Scientific Publishing, Singapore (2003)zbMATHGoogle Scholar
  21. 21.
    Yang, B.-Y., Chen, J.-M.: Theoretical Analysis of XL over Small Fields. In: Wang, H., Pieprzyk, J., Varadharajan, V. (eds.) ACISP 2004. LNCS, vol. 3108, pp. 277–288. Springer, Heidelberg (2004)CrossRefGoogle Scholar

Copyright information

© International Association for Cryptologic Research 2011

Authors and Affiliations

  • Jintai Ding
    • 1
    • 2
  • Timothy J. Hodges
    • 2
  1. 1.South China University of TechnologyGuangzhouChina
  2. 2.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA

Personalised recommendations