Book Chapter10.1007/3-540-11607-9_15
On Polynomial Factorization
Daniel Lazard
- 05 Apr 1982
- pp 126-134
13
TL;DR: It appears that the authors' and Cantor-Zassenhaus algorithms have the same asymptotic complexity and they are the best algorithms actually known ; with elementary multiplication and GCD computation, CantorZASSenhaus algorithm is always better than theirs ; with fast multiplication andGCD, it seems that theirs is better, but this fact is not yet proveen.
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Abstract: These algorithms are probabilistic in the following sense. The time of computation depends on random choices, but the validity of the result does not depend on them. So, worst case complexity, being infinite, is meaningless and we compute average complexity. It appears that our and Cantor-Zassenhaus algorithms have the same asymptotic complexity and they are the best algorithms actually known ; with elementary multiplication and GCD computation, CantorZassenhaus algorithm is always better than ours ; with fast multiplication and GCD, it seems that ours is better, but this fact is not yet proveen.
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Citations
Factorization of Polynomials
E. Kaltofen
- 01 Jan 1983
TL;DR: Algorithms for factoring polynomials in one or more variables over various coefficient domains are discussed and an attempt is made to establish a complete historic trace for today’s methods.
74
Decomposition of Algebras
Patrizia Gianni,Patrizia Gianni,Victor S. Miller,Barry M. Trager +3 more
- 04 Jul 1988
TL;DR: It is shown that it is possible to define a lifting process that allows to reconstruct the answer over the rational numbers and this lifting appears to be very efficient since it is a quadratic lifting that doesn't require stepwise inversions.
24
Algebraic algorithms in GF(q)
TL;DR: A module of algorithms implemented in the ALDES/SAC2 computer algebra system, which will be available with the next release of this system, on algebraic algorithms for computing in large Galois Fields GF(q), where p is the characteristic of the field and may be arbitrarily large.
14
Removing randomness from computational number theory
Victor Shoup,Eric Bach +1 more
- 01 Jan 1989
TL;DR: A deterministic polynomial time reduction from the latter problem to the former, giving rise to a deterministic algorithm for constructing irreducible polynomials that runs in polynometric time for elds of small characteristic, and a new deterministic factoring algorithm whose worst-case running time is asymptotically faster than that of previously known deterministic algorithms for this problem.
13
Univariate polynomial factorization over finite fields
Patrice Naudin,Claude Quitté +1 more
TL;DR: A detailed description of an efficient implementation of the Cantor-Zassenhaus algorithm used in the release 2 of the Axiom computer algebra system is given.
9
References
•Book
The Design and Analysis of Computer Algorithms
Alfred V. Aho,John E. Hopcroft +1 more
- 01 Jan 1974
TL;DR: This text introduces the basic data structures and programming techniques often used in efficient algorithms, and covers use of lists, push-down stacks, queues, trees, and graphs.
10.6K
Factoring polynomials over large finite fields
TL;DR: In this paper, the authors present a deterministic procedure for factoring polynomials over finite fields, which reduces the problem of factoring an arbitrary polynomial over the Galois field GF(p m) to finding the roots in GF(m) of certain other polynomorphisms over GF (m).
A new algorithm for factoring polynomials over finite fields
David G. Cantor,Hans Zassenhaus +1 more
TL;DR: A new probabilistic method is presented which, when combined with the above algorithms, avoids the need for both resultants and linear equations and leads to algorithms which are conceptually simpler than previous methods.
An inequality about factors of polynomials
TL;DR: In this paper, a sharp inequality about the product of some roots of a polynomial is proved, which is used to bound the height of the factors of a polynomial.
165
On the efficiency of algorithms for polynomial factoring
TL;DR: It is shown that if the characteristic of the field is of the form p = L 21 + 1, where I L, then the roots of a polynomial of degree n may be found in O(n log p + n log2 p) steps.