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Subquadratictime factoring of polynomials over finite fields
 Math. Comp
, 1998
"... 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 Θ(n 2+o(1)). The new algorithms rely on fast ..."
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Cited by 79 (11 self)
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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 Θ(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 Fq with q elements, the algorithms use O(n 1.815 log q) arithmetic operations in Fq. The new “baby step/giant step ” techniques used in our algorithms also yield new fast practical algorithms at superquadratic asymptotic running time, and subquadratictime methods for manipulating normal bases of finite fields. 1.
Searching for Primitive Roots in Finite Fields
, 1992
"... Let GF(p n ) be the finite field with p n elements where p is prime. We consider the problem of how to deterministically generate in polynomial time a subset of GF(p n ) that contains a primitive root, i.e., an element that generates the multiplicative group of nonzero elements in GF(p n ). ..."
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Cited by 50 (3 self)
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Let GF(p n ) be the finite field with p n elements where p is prime. We consider the problem of how to deterministically generate in polynomial time a subset of GF(p n ) that contains a primitive root, i.e., an element that generates the multiplicative group of nonzero elements in GF(p n ). We present three results. First, we present a solution to this problem for the case where p is small, i.e., p = n O(1) . Second, we present a solution to this problem under the assumption of the Extended Riemann Hypothesis (ERH) for the case where p is large and n = 2. Third, we give a quantitative improvement of a theorem of Wang on the least primitive root for GF(p) assuming the ERH. Appeared in Mathematics of Computation 58, pp. 369380, 1992. An earlier version of this paper appeared in the 22nd Annual ACM Symposium on Theory of Computing (1990), pp. 546554. 1980 Mathematics Subject Classification (1985 revision): 11T06. 1. Introduction Consider the problem of finding a primitive ...
Constructing nonresidues in finite fields and the extended Riemann hypothesis
 Math. Comp
, 1991
"... Abstract. We present a new deterministic algorithm for the problem of constructing kth power nonresidues in finite fields Fpn,wherepis prime and k is a prime divisor of pn −1. We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed n and p →∞, our algorithm runs in pol ..."
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Cited by 12 (0 self)
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Abstract. We present a new deterministic algorithm for the problem of constructing kth power nonresidues in finite fields Fpn,wherepis prime and k is a prime divisor of pn −1. We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed n and p →∞, our algorithm runs in polynomial time. Unlike other deterministic algorithms for this problem, this polynomialtime bound holds even if k is exponentially large. More generally, assuming the ERH, in time (n log p) O(n) we can construct a set of elements
Factoring Polynomials Over Finite Fields: A Survey
, 2001
"... This survey reviews several algorithms for the factorization of univariate polynomials over finite fields. We emphasize the main ideas of the methods and provide an uptodate bibliography of the problem. ..."
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Cited by 4 (1 self)
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This survey reviews several algorithms for the factorization of univariate polynomials over finite fields. We emphasize the main ideas of the methods and provide an uptodate bibliography of the problem.
Schemes for Deterministic Polynomial Factoring
, 2008
"... In this work we relate the deterministic complexity of factoring polynomials (over finite fields) to certain combinatorial objects we call mschemes. We extend the known conditional deterministic subexponential time polynomial factoring algorithm for finite fields to get an underlying mscheme. We d ..."
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In this work we relate the deterministic complexity of factoring polynomials (over finite fields) to certain combinatorial objects we call mschemes. We extend the known conditional deterministic subexponential time polynomial factoring algorithm for finite fields to get an underlying mscheme. We demonstrate how the properties of mschemes relate to improvements in the deterministic complexity of factoring polynomials over finite fields assuming the generalized Riemann Hypothesis (GRH). In particular, we give the first deterministic polynomial time algorithm (assuming GRH) to find a nontrivial factor of a polynomial of prime degree n where (n − 1) is a smooth number.
USING THE THEORY OF CYCLOTOMY TO FACTOR CYCLOTOMIC POLYNOMIALS OVER FINITE FIELDS
"... Abstract. We examine the problem of factoring the rth cyclotomic polynomial, Φr(x), over Fp, r and p distinct primes. Given the traces of the roots of Φr(x) weconstructthecoefficientsofΦr(x) intimeO(r 4). We demonstrate a deterministic algorithm for factoring Φr(x) intimeO((r 1/2+ɛ log p) 9)when Φr( ..."
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Abstract. We examine the problem of factoring the rth cyclotomic polynomial, Φr(x), over Fp, r and p distinct primes. Given the traces of the roots of Φr(x) weconstructthecoefficientsofΦr(x) intimeO(r 4). We demonstrate a deterministic algorithm for factoring Φr(x) intimeO((r 1/2+ɛ log p) 9)when Φr(x) has precisely two irreducible factors. Finally, we present a deterministic algorithm for computing the sum of the irreducible factors of Φr(x) intime O(r 6). 1.
Trading GRH for Algebra: Algorithms for Factoring Polynomials and Related Structures
, 2009
"... In this paper we develop techniques that eliminate the need of the Generalized Riemann Hypothesis (GRH) from various (almost all) known results about deterministic polynomial factoring over finite fields. Our main result shows that given a polynomial f(x) of degree n over a finite field k, we can fi ..."
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Cited by 2 (1 self)
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In this paper we develop techniques that eliminate the need of the Generalized Riemann Hypothesis (GRH) from various (almost all) known results about deterministic polynomial factoring over finite fields. Our main result shows that given a polynomial f(x) of degree n over a finite field k, we can find in deterministic poly(n log n, log k) time either a nontrivial factor of f(x) or a nontrivial automorphism of k[x]/(f(x)) of order n. This main tool leads to various new GRHfree results, most striking of which are: 1. Given a noncommutative algebra A of dimension n over a finite field k. There is a deterministic poly(n log n, log k) time algorithm to find a zero divisor in A. This is the best known deterministic GRHfree result since Friedl and Rónyai (STOC 1985) first studied the problem of finding zero divisors in finite algebras and showed that this problem has the same complexity as factoring polynomials over finite fields. 2. Given a positive integer r such that either 8r or r has at least two distinct odd prime factors. There is a deterministic polynomial time algorithm to find a nontrivial factor of the rth cyclotomic polynomial over a finite field. This is the best known deterministic GRHfree result since Huang (STOC 1985) showed that cyclotomic polynomials can be factored over finite fields in deterministic polynomial time assuming GRH. In this paper, following the seminal work of Lenstra (1991) on constructing isomorphisms between finite fields, we further generalize classical Galois theory constructs like cyclotomic extensions, Kummer extensions, Teichmüller subgroups, to the case of commutative semisimple algebras with automorphisms. These generalized constructs help eliminate the dependence on GRH.
FACTORING POLYNOMIALS AND GRÖBNER BASES
, 2009
"... Factoring polynomials is a central problem in computational algebra and number theory and is a basic routine in most computer algebra systems (e.g. Maple, Mathematica, Magma, etc). It has been extensively studied in the last few decades by many mathematicians and computer scientists. The main appro ..."
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Factoring polynomials is a central problem in computational algebra and number theory and is a basic routine in most computer algebra systems (e.g. Maple, Mathematica, Magma, etc). It has been extensively studied in the last few decades by many mathematicians and computer scientists. The main approaches include Berlekamp’s method (1967) based on the kernel of Frobenius map, Niederreiter’s method (1993) via an ordinary differential equation, Zassenhaus’s modular approach (1969), Lenstra, Lenstra and Lovasz’s lattice reduction (1982), and Gao’s method via a partial differential equation (2003). These methods and their recent improvements due to van Hoeij (2002) and Lecerf et al (2006– 2007) provide efficient algorithms that are widely used in practice today. This thesis studies two issues on polynomial factorization. One is to improve the efficiency of modular approach for factoring bivariate polynomials over finite fields. The usual modular approach first solves a modular linear equation (from Berlekamp’s equation or Niederreiter’s differential equation), then performs Hensel lifting of modular factors, and finally finds right combinations. An alternative method is presented in this thesis that