Solvability of a System of Bivariate Polynomial Equations over a Finite Field

(Extended Abstract)
  • Neeraj Kayal
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3580)


We investigate the complexity of the following polynomial solvability problem: Given a finite field \({\mathbb F}_{q}\) and a set of polynomials

$$f_{1}(x,y),f_{2}(x,y),...,f_{n}(x,y),g(x,y) \ \epsilon \ {\mathbb F}_{q} [x,y]$$

determine the \({\mathbb F}_{q}\)-solvability of the system

$$f_{1}(x,y)=f_{2}(x,y)=...=f_{n}(x,y)=0 \ {\rm and} \ {\it g}(x,y) \neq 0$$

We give a deterministic polynomial-time algorithm for this problem.


Deterministic Algorithm Irreducible Factor Factorization Algorithm Polynomial Factorization Permutation Polynomial 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [GKL04]
    Gao, S., Kaltofen, E., Lauder, A.: Deterministic distinc-degree factorization of polnomials over finite fields. Journal of Symbolic Computing 38(6), 1461–1470 (2004)CrossRefMathSciNetGoogle Scholar
  2. [HW99]
    Huang, M.-D., Wong, Y.-C.: Solvability of systems of polynomial congruences modulo a large prime. Computational Complexity 8(3), 227–257 (1999)MATHCrossRefMathSciNetGoogle Scholar
  3. [LM88]
    Lidl, R., Mullen, G.L.: When does a polynomial over a finite field permute the elements of the field? American Mathematical Monthly 95, 243–246 (1988)MATHCrossRefMathSciNetGoogle Scholar
  4. [LM93]
    Lidl, R., Mullen, G.L.: When does a polynomial over a finite field permute the elements of the field?, II. American Mathematical Monthly 100, 71–74 (1993)MATHCrossRefMathSciNetGoogle Scholar
  5. [LM83]
    Lidl, R., Muller, W.B.: Permutation Polynomials in RSA cryptosystems. In: Chaum, D. (ed.) Proceedings CRYPTO 1983, pp. 293–301 (1983)Google Scholar
  6. [MG94]
    Ma, K., Von Zur Gathen, J.: The computational complexity of recognizing permutation functions. Computational Complexity 5(1), 76–97 (1995)MATHCrossRefMathSciNetGoogle Scholar
  7. [Len05]
    Lenstra, H.: Private Communication (2005)Google Scholar
  8. [RSA78]
    Rivest, R.L., Shamir, A., Adleman, L.M.: A method for obtaining digital signatures and public-key cryptosystems. Communications of the ACM 21(2), 120–126 (1978)MATHCrossRefMathSciNetGoogle Scholar
  9. [Bac93]
    Bach, E.: Weil bounds for singular curves. Applicable Algebra in Engineering, Communication and Computing 7, 289–298 (1996)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Neeraj Kayal
    • 1
    • 2
  1. 1.Indian Institute of TechnologyKanpur
  2. 2.National University of Singapore 

Personalised recommendations