Factoring polynomials over arbitrary finite fields

Finite Fields and Applications, 2001

Results on the worst case behavior of the authors' extension (Theor. Comput. Sci. 234 (2000), 301-308) of Shoup's algorithm for factoring polynomials over finite prime fields (Inf. Process. Lett. 33 (1990), 261-267) are improved. Moreover, the consequences of the average case behavior of the extended algorithm for multivariate algorithms are described, and an extension of Lenstra's algorithm (Lond. Math. Soc, Lect. Note Ser. 154 (1990), 76-85) for root finding over finite prime fields is presented.

Electronic Colloquium on Computational Complexity, 2008

In this work we relate the deterministic complexity of factoring polynomials (over finite fields) to certain combinatorial objects, we call m-schemes, that are generalizations of permutation groups. We design a new generalization of the known conditional deterministic subexponential time polynomial factoring algorithm to get an underlying m-scheme. We then demonstrate how progress in understanding mschemes relate to improvements in the deterministic complexity of factoring polynomials, assuming the Generalized Riemann Hypothesis (GRH).

2011

Any non constant polynomial over a field can be expressed as a product of ir-reducible polynomials. In finite fields, some algorithms work for the calculation of irreducible factors of a polynomial of positive degree. The factorization of polynomials over finite fields has great ...

Journal de Théorie des Nombres de Bordeaux, 2009

We prove polynomial time complexity for a now widely used factorization algorithm for polynomials over the rationals. Our approach also yields polynomial time complexity results for bivariate polynomials over a finite field.

Journal of Symbolic Computation, 2005

In this paper we present a generic algorithm for factoring polynomials over global fields F. As efficient implementations of that algorithm for number fields and function fields differ substantially, these cases will be treated separately. Complexity issues and implementations will be discussed in part II which also contains illustrative examples.

Theoretical Computer Science, 1997

In this paper we present a new deterministic algorithm for computing the square-free decomposition of multivariate polynomials with coefficients from a finite field. Our algorithm is based on Yun's square-free factorization algorithm for characteristic 0. The new algorithm is more efficient than existing, deterministic algorithms based on Musser's squarefree algorithm. We will show that the modular approach presented by Yun has no significant performance advantage over our algorithm. The new algorithm is also simpler to implement and it can rely on any existing GCD algorithm without having to worry about choosing "good" evaluation points. To demonstrate this, we present some timings using implementations in Maple (Char et al., 1991), where the new algorithm is used for Release 4 onwards, and Axiom (Jenks and Sutor, 1992) which is the only system known to the author to use an implementation of Yun's modular algorithm mentioned above.

Proceedings of the 2009 international symposium on Symbolic and algebraic computation - ISSAC '09, 2009

We present an efficient algorithm for factoring a multivariate polynomial f ∈ L[x1,. .. , xv] where L is an algebraic function field with k ≥ 0 parameters t1,. .. , t k and r ≥ 0 field extensions. Our algorithm uses Hensel lifting and extends the EEZ algorithm of Wang which was designed for factorization over Q. We also give a multivariate p-adic lifting algorithm which uses sparse interpolation. This enables us to avoid using poor bounds on the size of the integer coefficients in the factorization of f when using Hensel lifting. We have implemented our algorithm in Maple 13. We provide timings demonstrating the efficiency of our algorithm.

Journal of Symbolic Computation, 2005

In this paper we describe software for an efficient factorization of polynomials over global fields F. The algorithm for function fields was recently incorporated into our system KANT. The method is based on a generic algorithm developed by the second author in an earlier paper in this journal. Besides algorithmic aspects not contained in that paper we give details about the current implementation and about some complexity issues as well as a few illustrative examples. Also, a generalization of the application of LLL reduction for factoring polynomials over arbitrary global fields is developed.

2006

Self-reciprocal irreducible monic (srim) polyn omials over finite fields have been studied in the past. These polynomials can be studied in the context of quad ratic transformation of irreducible polynomials over finite fields. In this talk we present the generalization of some of the results known about srim polynomials to polynomials obtained by quadratic transformation of irreducible polynomials over finite fields. Speaker:Dan Bernstein (University of Illinois at Chicago) Title: Faster factorization into coprimes Abstract: How quickly can we factor a set of univariate polyn mials into coprimes? See http://cr.yp.to/coprimes.html for examples and applications. Bach, Driscoll, and Shallit chieved time n in 1990, wheren is the number of input coefficients; I achieved time n(lg n) in 1995; much more recently I achieved time n(lg n). Speaker:Antonia Bluher (National Security Agency) Title: Hyperquadratic elements of degree 4 Abstract: I will describe joint work with Alain Lasjaunias a ...

State of the art factoring in Q[x] is dominated in theory by a combinatorial reconstruction problem while, excluding some rare polynomials, performance tends to be dominated by Hensel lifting. We present an algorithm which gives a practical improvement (less Hensel lifting) for these more common polynomials. In addition, factoring has suffered from a 25 year complexity gap because the best implementations are much faster in practice than their complexity bounds. We illustrate that this complexity gap can be closed by providing an implementation which is comparable to the best current implementations and for which competitive complexity results can be proved.