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Factoring Polynomials over Finite Fields

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Applications of Finite Fields

Part of the book series: The Springer International Series in Engineering and Computer Science ((SECS,volume 199))

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Abstract

A polynomial of degree n over a finite field F q is an expression in an indeterminate x having the form

$$f(x) = \sum\limits_{i = 0}^n {{a_i}{x^1}} $$

where n is a non-negative integer, a i ∈ F q , 0 ≤ i ≤ n and a n ≠ 0. To be more precise, f (x) is called a univariate polynomial to distinguish the more general situation where more indeterminates are involved. Most of this chapter will deal with univariate polynomials but the multivariate case will be briefly mentioned at the end.

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References

  1. M. Ben-Or, “Probabilistic algorithms in finite fields”, 22nd Annual Symposium on Foundations of Computer Science (1981), 394–398.

    Google Scholar 

  2. E.R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, New York, 1968.

    MATH  Google Scholar 

  3. E.R. Berlekamp, “Factoring polynomials over large finite fields”, Math. Comp., 24 (1970), 713–735.

    Article  MathSciNet  Google Scholar 

  4. E.R. Berlekamp, H. Rumsey and G. Solomon, “On the solution of algebraic equations over finite fields”, Information and Control, 10 (1967), 553–564.

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Camion, “A deterministic algorithm for factoring polynomials of F q [x]” , Annals of Discrete Math., 17 (1983), 149–157.

    Google Scholar 

  6. P. Camion, “Improving an algorithm for factoring polynomials over a finite field and constructing large irreducible polynomials”, IEEE Trans. Info. Th., 29 (1983), 378–385.

    Article  MathSciNet  MATH  Google Scholar 

  7. D. Cantor and H. Zassenhaus, “A new algorithm for factoring polynomials over finite fields” , Math. Comp., 36 (1981), 587–592.

    Article  MathSciNet  MATH  Google Scholar 

  8. B. Chor and R. Rivest, “A knapsack-type public key cryptosystem based on arithmetic in finite fields”, IEEE Trans. Info. Th., 34 (1988), 901–909.

    Article  MathSciNet  Google Scholar 

  9. J. Von Zur Gathen, “Irreducibility of multivariate polynomials”, J. Comput. System Sci., 31 (1985), 225–264.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Von Zur Gathen, “Factoring polynomials and primitive elements for special primes”, Theoretical Computer Science, 52 (1987), 77–89.

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Von Zur Gathen and E. Kaltofen, “Factoring sparse multivariate polynomials”, J. Comput. System Sci., 31 (1985), 265–287.

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Von Zur Gathen and E. Kaltofen, “Factorization of multivariate polynomials over finite fields” , Math. Comp., 45 (1985), 251–261.

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Huang, “Riemann hypothesis and finding roots over finite fields”, Proceedings of the 17th Annual Acm Symposium on Theory of Computing (1985), 121–130.

    Google Scholar 

  14. A. Lenstra, “Factoring multivariate polynomials over finite fields”, J. Comput. System Sci., 30 (1985), 235–248.

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Lenstra, H.W. Lenstra and L. Lovasz, “Factoring polynomials with rational coefficients”, Math. Ann., 261 (1982), 515–534.

    Article  MathSciNet  MATH  Google Scholar 

  16. H.W. Lenstra, “On the Chor-Rivest knapsack cryptosystem”, J. of Cryptology, 3 (1991), 149–155.

    Article  MathSciNet  MATH  Google Scholar 

  17. F.J. Macwilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, North-HollAnd, Amsterdam, 1977 .

    MATH  Google Scholar 

  18. A. Menezes, P. Van Oorschot and S. Vanstone, “Some computational aspects of root finding in GF(g m ) n ”, in Symbolic and Algebraic Computation, Lecture Notes in Computer Science, 358 (1989), 259–270.

    Article  Google Scholar 

  19. M. Rabin, “Probabilistic algorithms in finite fields”, SIAM J. Comput., 9 (1980), 273–280.

    Article  MathSciNet  MATH  Google Scholar 

  20. L. Rónyai, “Factoring polynomials over finite fields”, J. of Algorithms, 9 (1988), 391–400.

    Article  MATH  Google Scholar 

  21. L. Rónyai, “Factoring polynomials modulo special primes”, Combinatorica, 9 (1989), 199–206.

    Article  MathSciNet  MATH  Google Scholar 

  22. V. Shoup, “On the deterministic complexity of factoring polynomials over finite fields”, Information Processing Letters, 33 (1990), 261–267.

    Article  MathSciNet  MATH  Google Scholar 

  23. V. Shoup, “Smoothness and factoring polynomials over finite fields”, Information Processing Letters, 38 (1991), 39–42.

    Article  MathSciNet  MATH  Google Scholar 

  24. I.E. Shparlinski, Computational Problems in Finite Fields, Kluwer Academic Publishers, 1992.

    Book  MATH  Google Scholar 

  25. P. Van Oorschot and S. Vanstone, “A geometric approach to root finding in GF(g m ) n ”, IEEE Trans. Info. Th., 35 (1989), 444–453.

    Article  MATH  Google Scholar 

  26. D. Wan, “Factoring multivariate polynomials over large finite fields”, Math. Comp., 54 (1990), 755–770.

    Article  MathSciNet  MATH  Google Scholar 

  27. H. Zassenhaus, “On Hensel factorization I”, J. Number Theory, 1 (1969), 291–311.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. University of Waterloo, Canada

    Ian F. Blake, XuHong Gao, Ronald C. Mullin, Scott A. Vanstone & Tomik Yaghoobian

Authors
  1. Ian F. Blake
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  2. XuHong Gao
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  3. Ronald C. Mullin
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  4. Scott A. Vanstone
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  5. Tomik Yaghoobian
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Editor information

Alfred J. Menezes

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© 1993 Springer Science+Business Media New York

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Blake, I.F., Gao, X., Mullin, R.C., Vanstone, S.A., Yaghoobian, T. (1993). Factoring Polynomials over Finite Fields. In: Menezes, A.J. (eds) Applications of Finite Fields. The Springer International Series in Engineering and Computer Science, vol 199. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-2226-0_2

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  • DOI: https://doi.org/10.1007/978-1-4757-2226-0_2

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-5130-4

  • Online ISBN: 978-1-4757-2226-0

  • eBook Packages: Springer Book Archive

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