Designs, Codes and Cryptography

, Volume 49, Issue 1–3, pp 47–60 | Cite as

On solving sparse algebraic equations over finite fields

  • Igor Semaev


A system of algebraic equations over a finite field is called sparse if each equation depends on a small number of variables. In this paper new deterministic algorithms for solving such equations are presented. The mathematical expectation of their running time is estimated. These estimates are at present the best theoretical bounds on the complexity of solving average instances of the above problem. In characteristic 2 the estimates are significantly lower the worst case bounds provided by SAT solvers.


Sparse algebraic equations over finite fields Constraint satisfaction problem Gluing Random allocations 

AMS Classifications

11T71 68Q25 


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  1. 1.
    Bardet M., Faugére J.-C., Salvy B.: Complexity of Gröbner basis computation for semiregular overdetermined sequences over F2 with solutions in F2. Research report RR–5049, INRIA (2003).Google Scholar
  2. 2.
    Balakin G.V., Bachurin S.A.: Evaluation of the successive search for unknowns, (in Russian), Trudy po diskretnoj matematike, vol. 6, pp. 7–13. Fizmatlit (2002).Google Scholar
  3. 3.
    Courtois N., Klimov A., Patarin J., Shamir A.: Efficient algorithms for solving overdefined systems of multivariate polynomial equations. In: Eurocrypt 2000, LNCS 1807, pp. 392–407. Springer-Verlag (2000).Google Scholar
  4. 4.
    Chistyakov V.P. (1967). Discrete limit distributions in the problem of shots with arbitrary probabilities of occupancy of boxes. Matem. Zametki 1: 9–16MATHGoogle Scholar
  5. 5.
    Faugére J.-C.: A new efficient algorithm for computing Gröbner bases without reduction to zero (F5). In: Proc. ISSAC 2002, pp. 75–83. ACM Press (2002).Google Scholar
  6. 6.
    Iwama K. (2004). Worst-case upper bounds for kSAT. Bull. EATCS 82: 61–71MathSciNetGoogle Scholar
  7. 7.
    Kolchin V. (1966). The rate of convergence to limit distributions in the classical problem of shots. Teoriya veroyatn. i yeye primenen. 11: 144–156Google Scholar
  8. 8.
    Kolchin V., Sevast’yanov A., Chistyakov V.: Random Allocations. Wiley (1978).Google Scholar
  9. 9.
    Raddum H.: Solving non-linear sparse equation systems over GF(2) using graphs. University of Bergen, preprint, 2004.Google Scholar
  10. 10.
    Raddum H., Semaev I.: New technique for solving sparse equation systems. Cryptology ePrint Archive, 2006/475.Google Scholar
  11. 11.
    Raddum H., Semaev I.: Solving MRHS linear equations. Extended abstract, accepted at WCC (2007).Google Scholar
  12. 12.
    Tsang E.P.K.: Foundations of Constraint Satisfaction. Academic Press (1993).Google Scholar
  13. 13.
    Yang B.-Y., Chen J.-M., Courtois N.: On asymptotic security estimates in XL and Gröbner bases-related algebraic cryptanalysis. In: ICICS 2004, LNCS 3269, pp. 401–413. Springer-Verlag (2004).Google Scholar
  14. 14.
    Zakrevskij A., Vasilkova I.: Reducing large systems of Boolean equations. In: 4th International Workshop on Boolean Problems, Freiberg University, September, 21–22 (2000).Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of InformaticsUniversity of BergenBergenNorway

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