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)

Abstract

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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