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research-article

Factoring Polynomials over Special Finite Fields

Authors:

Joachim von zur Gathen,

Hendrik W. LenstraAuthors Info & Claims

Finite Fields and Their Applications, Volume 7, Issue 1

Pages 5 - 28

https://doi.org/10.1006/ffta.2000.0306

Published: 01 January 2001 Publication History

Abstract

We exhibit a deterministic algorithm for factoring polynomials in one variable over finite fields. It is efficient only if a positive integer k is known for which k(p) is built up from small prime factors; here k denotes the kth cyclotomic polynomial, and p is the characteristic of the field. In the case k=1, when k(p)=p 1, such an algorithm was known, and its analysis required the generalized Riemann hypothesis. Our algorithm depends on a similar, but weaker, assumption; specifically, the algorithm requires the availability of an irreducible polynomial of degree r over Z/pZ for each prime number r for which k(p) has a prime factor l with l 1 mod r. An auxiliary procedure is devoted to the construction of roots of unity by means of Gauss sums. We do not claim that our algorithm has any practical value.

References

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L. M. Adleman and H. W. Lenstra, Jr., Finding irreducible polynomials over finite fields, inProceedings of the 18th Annual ACM Symposium on Theory of Computing (STOC), Berkeley, 1986, pp. 350-355.

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Factorization of polynomials over finite fields in subexponential time under GRH. In: Adleman, L.M., Huang, M.-D. (Eds.), Algorithmic Number Theory (ANTS-I), 877. Springer¿Verlag, Berlin. pp. 209-219.

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Factoring polynomials and primitive elements for special primes. Theoret. Comput. Sci. v52. 77-89.

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Cited By

Grześkowiak M(2019)Algorithms for Relatively Cyclotomic PrimesFundamenta Informaticae10.5555/2594934.2594938125:2(161-181)Online publication date: 4-Jan-2019
https://dl.acm.org/doi/10.5555/2594934.2594938
Bhargava VIvanyos GMittal RSaxena NYap CEl Din M(2017)Irreducibility and Deterministic r-th Root Finding over Finite FieldsProceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3087604.3087620(37-44)Online publication date: 23-Jul-2017
https://dl.acm.org/doi/10.1145/3087604.3087620
Ivanyos GKarpinski MSaxena NJohnson JPark HKaltofen E(2009)Schemes for deterministic polynomial factoringProceedings of the 2009 international symposium on Symbolic and algebraic computation10.1145/1576702.1576730(191-198)Online publication date: 28-Jul-2009
https://dl.acm.org/doi/10.1145/1576702.1576730
Show More Cited By

Factoring Polynomials over Special Finite Fields
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis

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Published In

cover image Finite Fields and Their Applications

Finite Fields and Their Applications Volume 7, Issue 1

January, 2001

252 pages

ISSN:1071-5797

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Copyright © Academic Press.

Publisher

Elsevier Science Publishers B. V.

Netherlands

Publication History

Published: 01 January 2001

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Cited By

Grześkowiak M(2019)Algorithms for Relatively Cyclotomic PrimesFundamenta Informaticae10.5555/2594934.2594938125:2(161-181)Online publication date: 4-Jan-2019
https://dl.acm.org/doi/10.5555/2594934.2594938
Bhargava VIvanyos GMittal RSaxena NYap CEl Din M(2017)Irreducibility and Deterministic r-th Root Finding over Finite FieldsProceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3087604.3087620(37-44)Online publication date: 23-Jul-2017
https://dl.acm.org/doi/10.1145/3087604.3087620
Ivanyos GKarpinski MSaxena NJohnson JPark HKaltofen E(2009)Schemes for deterministic polynomial factoringProceedings of the 2009 international symposium on Symbolic and algebraic computation10.1145/1576702.1576730(191-198)Online publication date: 28-Jul-2009
https://dl.acm.org/doi/10.1145/1576702.1576730
Maurer URaub D(2007)Black-box extension fields and the inexistence of field-homomorphic one-way permutationsProceedings of the Advances in Crypotology 13th international conference on Theory and application of cryptology and information security10.5555/1781454.1781490(427-443)Online publication date: 2-Dec-2007
https://dl.acm.org/doi/10.5555/1781454.1781490
van de Woestijne CGao XLabahn G(2005)Deterministic equation solving over finite fieldsProceedings of the 2005 international symposium on Symbolic and algebraic computation10.1145/1073884.1073932(348-353)Online publication date: 24-Jul-2005
https://dl.acm.org/doi/10.1145/1073884.1073932

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