Nothing Special   »   [go: up one dir, main page]

Skip to main content

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

(Extended Abstract)

  • Conference paper
Automata, Languages and Programming (ICALP 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3580))

Included in the following conference series:

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Gao, S., Kaltofen, E., Lauder, A.: Deterministic distinc-degree factorization of polnomials over finite fields. Journal of Symbolic Computing 38(6), 1461–1470 (2004)

    Article  MathSciNet  Google Scholar 

  2. Huang, M.-D., Wong, Y.-C.: Solvability of systems of polynomial congruences modulo a large prime. Computational Complexity 8(3), 227–257 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. 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)

    Article  MATH  MathSciNet  Google Scholar 

  4. 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)

    Article  MATH  MathSciNet  Google Scholar 

  5. Lidl, R., Muller, W.B.: Permutation Polynomials in RSA cryptosystems. In: Chaum, D. (ed.) Proceedings CRYPTO 1983, pp. 293–301 (1983)

    Google Scholar 

  6. Ma, K., Von Zur Gathen, J.: The computational complexity of recognizing permutation functions. Computational Complexity 5(1), 76–97 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  7. Lenstra, H.: Private Communication (2005)

    Google Scholar 

  8. 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)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bach, E.: Weil bounds for singular curves. Applicable Algebra in Engineering, Communication and Computing 7, 289–298 (1996)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kayal, N. (2005). Solvability of a System of Bivariate Polynomial Equations over a Finite Field. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds) Automata, Languages and Programming. ICALP 2005. Lecture Notes in Computer Science, vol 3580. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11523468_45

Download citation

  • DOI: https://doi.org/10.1007/11523468_45

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-27580-0

  • Online ISBN: 978-3-540-31691-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics