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Mathematics of Computation

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Subquadratic-time factoring of polynomials
over finite fields

Authors: Erich Kaltofen and Victor Shoup
Journal: Math. Comp. 67 (1998), 1179-1197
MSC (1991): Primary 12E20, 13P05, 68Q40
MathSciNet review: 1459389
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Abstract: New probabilistic algorithms are presented for factoring univariate polynomials over finite fields. The algorithms factor a polynomial of degree $n$ over a finite field of constant cardinality in time $O(n^{1.815})$. Previous algorithms required time $\Theta(n^{2+o(1)})$. The new algorithms rely on fast matrix multiplication techniques. More generally, to factor a polynomial of degree $n$ over the finite field ${\mathbb F}_q$ with $q$ elements, the algorithms use $O(n^{1.815} \log q)$ arithmetic operations in ${\mathbb F}_q$.

The new ``baby step/giant step'' techniques used in our algorithms also yield new fast practical algorithms at super-quadratic asymptotic running time, and subquadratic-time methods for manipulating normal bases of finite fields.

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Additional Information

Erich Kaltofen
Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205

Victor Shoup
Affiliation: Bellcore, 445 South St., Morristown, New Jersey 07960-6438
Address at time of publication: IBM Zurich Research Laboratory, Säumerstrasse 4, Ch-8803 Rüschlikon, Switzerland

Keywords: Factoring, polynomials, finite fields, randomized algorithms, normal bases
Received by editor(s): October 12, 1995
Received by editor(s) in revised form: March 29, 1996
Additional Notes: This material is based on work supported in part by the National Science Foundation under Grant No. CCR-9319776 (first author) and by an Alexander von Humboldt Research Fellowship (second author).
A major part of the work was performed while the first author was at Rensselaer Polytechnic Institute, Department of Computer Science, in Troy, New York and while the second author was at the Universität des Saarlandes, FB 14–Informatik, in Saarbrücken, Germany.
A preliminary version of this paper appears in the Proc. 27th Annual ACM Symp. Theory of Computing, ACM Press, pp. 398–406 (1995).
Article copyright: © Copyright 1998 American Mathematical Society