Skip to main content
Log in

Probabilistic algorithm for finding roots of linearized polynomials

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

A probabilistic algorithm is presented for finding a basis of the root space of a linearized polynomial

$$L(x) = \sum_{i=0}^t L_i x^{q^i}$$

over \(\mathbb {F}_{q^n}\) . The expected time complexity of the presented algorithm is O(n t) operations in \(\mathbb {F}_{q^n}\) .

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Aho A.V., Hopcroft J.E. and Ullman J.D. (1974). The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading, MA

    MATH  Google Scholar 

  2. Augot D., Charpin P. and Sendrier N. (1992). Studying the locator polynomials of minimum weight codewords of BCH codes. IEEE Trans. Inform. Theory 38: 960–973

    Article  MATH  MathSciNet  Google Scholar 

  3. Ben-Or, M.: Probabilistic algorithms in finite fields. In: Proceedings of 22nd Annual IEEE Symposium Foundations of Computer Science (FOCS’1981). Nashville, TN pp. 394–398 (1981).

  4. Coppersmith D. and Winograd S. (1990). Matrix multiplication via arithmetic progressions. J. Symb. Comput. 9: 251–280

    Article  MATH  MathSciNet  Google Scholar 

  5. Delsarte P. (1978). Bilinear forms over a finite field, with applications to coding theory. J. Comb. Theory A 25: 226–241

    Article  MATH  MathSciNet  Google Scholar 

  6. Gabidulin E.M. (1985). Theory of codes with maximum rank distance. Probl. Inform. Transm. 21: 1–12

    MATH  Google Scholar 

  7. Gao S., von zur Gathen J., Panario D. and Shoup V. (2000). Algorithms for exponentiation in finite fields. J. Symb. Comput. 29: 879–889

    Article  MATH  Google Scholar 

  8. von zur Gathen J. and Gerhard J. (1999). Modern Computer Algebra. Cambridge University Press, Cambridge, UK

    MATH  Google Scholar 

  9. Lidl R. and Niederreiter H. (1997). Finite Fields, 2nd ed. Cambridge University Press, Cambridge, UK

    Google Scholar 

  10. Loidreau, P.: A Welch–Berlekamp like algorithm for decoding Gabidulin codes. In: Proceedings of the 4th International Workshop on Coding and Cryptography (WCC’2005). Bergen, Norway pp. 36–45 (2005).

  11. MacWilliams F.J. and Sloane N.J.A. (1977). The Theory of Error-Correcting Codes. North-Holland, Amsterdam, The Netherlands

    MATH  Google Scholar 

  12. Ore O. (1933). On a special class of polynomials. Trans. Amer. Math. Soc. 35: 559–584

    Article  MATH  MathSciNet  Google Scholar 

  13. Ore O. (1934). Contributions to the theory of finite fields. Trans. Amer. Math. Soc. 36: 243–274

    Article  MATH  MathSciNet  Google Scholar 

  14. Rabin M.O. (1980). Probabilistic algorithms in finite fields. SIAM J. Comput. 9: 273–280

    Article  MATH  MathSciNet  Google Scholar 

  15. Roth R.M. (1991). Maximum-rank array codes and their application to crisscross error correction. IEEE Trans. Inform. Theory 37: 328–336

    Article  MATH  MathSciNet  Google Scholar 

  16. Roth R.M. (1997). Probabilistic crisscross error correction. IEEE Trans. Inform. Theory 43: 1425–1436

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Vitaly Skachek.

Additional information

Communicated by S. Gao.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Skachek, V., Roth, R.M. Probabilistic algorithm for finding roots of linearized polynomials. Des. Codes Cryptogr. 46, 17–23 (2008). https://doi.org/10.1007/s10623-007-9125-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-007-9125-y

Keywords

AMS Classifications

Navigation