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54
Factoring polynomials with rational coefficients
 MATH. ANN
, 1982
"... In this paper we present a polynomialtime algorithm to solve the following problem: given a nonzero polynomial fe Q[X] in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q[X]. It is well known that this is equivalent to factoring primitive polynomia ..."
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Cited by 982 (11 self)
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In this paper we present a polynomialtime algorithm to solve the following problem: given a nonzero polynomial fe Q[X] in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q[X]. It is well known that this is equivalent to factoring primitive polynomials feZ[X] into irreducible factors in Z[X]. Here we call f ~ Z[X] primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. [8]. Its running time, measured in bit operations, is O(nl2+n9(log[fD3). Here f~Tl[X] is the polynomial to be factored, n = deg(f) is the degree of f, and for a polynomial ~ a ~ i with real coefficients a i. i An outline of the algorithm is as follows. First we find, for a suitable small prime number p, a padic irreducible factor h of f, to a certain precision. This is done with Berlekamp's algorithm for factoring polynomials over small finite fields, combined with Hensel's lemma. Next we look for the irreducible factor h o of f in
Factoring Polynomials and the Knapsack Problem.
"... Although a polynomial time algorithm exists, the most commonly used algorithm for factoring a univariate polynomial f with integer coefficients is the BerlekampZassenhaus algorithm which has a complexity that depends exponentially on n where n is the number of modular factors of f . This expone ..."
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Cited by 56 (17 self)
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Although a polynomial time algorithm exists, the most commonly used algorithm for factoring a univariate polynomial f with integer coefficients is the BerlekampZassenhaus algorithm which has a complexity that depends exponentially on n where n is the number of modular factors of f . This exponential time complexity is due to a combinatorial problem; the problem of choosing the right subset of these n factors. In this paper we reduce this combinatorial problem to a knapsack problem of a kind that can be solved with polynomial time algorithms such LLL or PSLQ. The result is a practical algorithm that can factor large polynomials even when n is large as well. 1 Introduction Let f be a polynomial of degree N with integer coefficients, f = N X i=0 a i x i where a i 2 ZZ. Assume that f is monic (i.e. aN = 1) and that f is squarefree (no multiple roots), so the gcd of f and f 0 equals 1. Let p be a prime number and let F p = ZZ=(p) be the field with p elements. Let ZZ p ...
On the Periods of Generalized Fibonacci Recurrences
, 1992
"... We give a simple condition for a linear recurrence (mod 2 w ) of degree r to have the maximal possible period 2 w 1 (2 r 1). It follows that the period is maximal in the cases of interest for pseudorandom number generation, i.e. for 3term linear recurrences dened by trinomials which are prim ..."
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Cited by 35 (11 self)
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We give a simple condition for a linear recurrence (mod 2 w ) of degree r to have the maximal possible period 2 w 1 (2 r 1). It follows that the period is maximal in the cases of interest for pseudorandom number generation, i.e. for 3term linear recurrences dened by trinomials which are primitive (mod 2) and of degree r > 2. We consider the enumeration of certain exceptional polynomials which do not give maximal period, and list all such polynomials of degree less than 15. 1.
Factorization of Polynomials Given by StraightLine Programs
 Randomness and Computation
, 1989
"... An algorithm is developed for the factorization of a multivariate polynomial represented by traightline program into its irreducible factors. The algorithm is in random polynomialtime as a function in the input size, total degree, and binary coefficient length for the usual coefficient fields and ..."
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Cited by 35 (8 self)
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An algorithm is developed for the factorization of a multivariate polynomial represented by traightline program into its irreducible factors. The algorithm is in random polynomialtime as a function in the input size, total degree, and binary coefficient length for the usual coefficient fields and outputs a straightline program, which with controllably high probability correctly determines the irreducible factors. It also returns the probably correct multiplicities of each distinct factor. If th oefficient field has finite characteristic p and p divides the multiplicities of some irreducible factors our algorithm constructs straightline programs for the appropriate pth powers of such factors. Also a probabilistic algorithm is presented that allows to convert a polynomial given by a straightline program into its sparse representation. This conversion algorithm is in randompolynomial time in the previously cited parameters and in an upper bound for the number of nonzero...
A relative van Hoeij algorithm over number fields
 J. Symbolic Computation
, 2004
"... Abstract. Van Hoeij’s algorithm for factoring univariate polynomials over the rational integers rests on the same principle as BerlekampZassenhaus, but uses lattice basis reduction to improve drastically on the recombination phase. His ideas give rise to a collection of algorithms, differing greatl ..."
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Cited by 24 (1 self)
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Abstract. Van Hoeij’s algorithm for factoring univariate polynomials over the rational integers rests on the same principle as BerlekampZassenhaus, but uses lattice basis reduction to improve drastically on the recombination phase. His ideas give rise to a collection of algorithms, differing greatly in their efficiency. We present two deterministic variants, one of which achieves excellent overall performance. We then generalize these ideas to factor polynomials over
Lifting and recombination techniques for absolute factorization
 J. Complexity
, 2007
"... Abstract. In the vein of recent algorithmic advances in polynomial factorization based on lifting and recombination techniques, we present new faster algorithms for computing the absolute factorization of a bivariate polynomial. The running time of our probabilistic algorithm is less than quadratic ..."
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Cited by 23 (7 self)
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Abstract. In the vein of recent algorithmic advances in polynomial factorization based on lifting and recombination techniques, we present new faster algorithms for computing the absolute factorization of a bivariate polynomial. The running time of our probabilistic algorithm is less than quadratic in the dense size of the polynomial to be factored.
Improved dense multivariate polynomial factorization algorithms
 J. Symbolic Comput
, 2005
"... We present new deterministic and probabilistic algorithms that reduce the factorization of dense polynomials from several to one variable. The deterministic algorithm runs in subquadratic time in the dense size of the input polynomial, and the probabilistic algorithm is softly optimal when the numb ..."
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Cited by 19 (3 self)
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We present new deterministic and probabilistic algorithms that reduce the factorization of dense polynomials from several to one variable. The deterministic algorithm runs in subquadratic time in the dense size of the input polynomial, and the probabilistic algorithm is softly optimal when the number of variables is at least three. We also investigate the reduction from several to two variables and improve the quantitative version of Bertini’s irreducibility theorem. Key words: Polynomial factorization, Hensel lifting, Bertini’s irreducibility theorem.
Sharp precision in Hensel lifting for bivariate polynomial factorization
 Mathematics of Computation
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Factoring bivariate sparse (lacunary) polynomials
 J. Complexity
"... Abstract. We present a deterministic algorithm for computing all irreducible factors of degree ≤ d of a given bivariate polynomial f ∈ K[x,y] over an algebraic number field K and their multiplicities, whose running time is polynomial in the bit length of the sparse encoding of the input and in d. Mo ..."
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Cited by 13 (1 self)
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Abstract. We present a deterministic algorithm for computing all irreducible factors of degree ≤ d of a given bivariate polynomial f ∈ K[x,y] over an algebraic number field K and their multiplicities, whose running time is polynomial in the bit length of the sparse encoding of the input and in d. Moreover, we show that the factors over Q of degree ≤ d which are not binomials can also be computed in time polynomial in the sparse length of the input and in d.
Factoring Univariate Polynomials over the Rationals
"... This thesis presents an algorithm for factoring polynomials over the rationals which follows the approach of the van Hoeij algorithm. The key theoretical novelty in our approach is that it is set up in a way that will make it possible to prove a new complexity result for this algorithm which was act ..."
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Cited by 12 (5 self)
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This thesis presents an algorithm for factoring polynomials over the rationals which follows the approach of the van Hoeij algorithm. The key theoretical novelty in our approach is that it is set up in a way that will make it possible to prove a new complexity result for this algorithm which was actually observed on prior algorithms. One difference of this algorithm from prior algorithms is the practical improvement which we call early termination. Our algorithm should outperform prior algorithms in many common classes of polynomials (including