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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 68 (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 40 (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 ...
A Fast Deterministic Algorithm for Factoring Polynomials over Finite Fields of Small Characteristic
 in Proc. 1991 International Symposium on Symbolic and Algebraic Computation
, 1991
"... We present a new algorithm for factoring polynomials over finite fields. Our algorithm is deterministic, and its running time is "almost" quadratic when the characteristic is a small fixed prime. As such, our algorithm is asymptotically faster than previously known deterministic algorithms for facto ..."
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Cited by 18 (2 self)
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We present a new algorithm for factoring polynomials over finite fields. Our algorithm is deterministic, and its running time is "almost" quadratic when the characteristic is a small fixed prime. As such, our algorithm is asymptotically faster than previously known deterministic algorithms for factoring polynomials over finite fields of small characteristic. Appeared in Proc. 1991 International Symposium on Symbolic and Algebraic Computation (ISSAC), pp. 1421, 1991. 1. Introduction Consider the problem of factoring a univariate polynomial f of degree n over the finite field F q , where q = p k and p is a small, fixed prime. We assume that F q is represented as F p (`), where ` is the root of an irreducible polynomial over F p of degree k. We present a new deterministic algorithm for this problem whose asymptotic complexity is less than that of previous deterministic algorithms. In discussing running times of algorithms, for expositional purposes we treat p as a constant in Sec...
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 8 (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
Removing Randomness From Computational Number Theory
, 1989
"... In recent years, many probabilistic algorithms (i.e., algorithms that can toss coins) that run in polynomial time have been discovered for problems with no known deterministic polynomial time algorithms. Perhaps the most famous example is the problem of testing large (say, 100 digit) numbers for pri ..."
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Cited by 3 (1 self)
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In recent years, many probabilistic algorithms (i.e., algorithms that can toss coins) that run in polynomial time have been discovered for problems with no known deterministic polynomial time algorithms. Perhaps the most famous example is the problem of testing large (say, 100 digit) numbers for primality. Even for problems which are known to have deterministic polynomial time algorithms, these algorithms are often not as fast as some probabilistic algorithms for the same problem. Even though probabilistic algorithms are useful in practice, we would like to know, for both theoretical and practical reasons, if randomization is really necessary to obtain the most efficient algorithms for certain problems. That is, we would like to know for which problems there is an inherent gap between the deterministic and probabilistic complexities of these problems. In this research, we consider two problems of a number theoretic nature: factoring polynomials over finite fields and constructing irred...
Trading GRH for Algebra: Algorithms for Factoring Polynomials and Related Structures
, 811
"... 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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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. 1
On the Complexity of Certain Algebraic and Number Theoretic Problems
"... It is certified that the work contained in the thesis entitled “On the Complexity ..."
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It is certified that the work contained in the thesis entitled “On the Complexity
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 smoothness of p − 1 for computing roots modulo p
, 2008
"... We prove, without recourse to the Extended Riemann Hypothesis, that the projection modulo p of any prefixed polynomial with integer coefficients can be completely factored in deterministic polynomial time if p − 1 has a (ln p) O(1)smooth divisor exceeding (p − 1) 1 2 +δ for some arbitrary small δ. ..."
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We prove, without recourse to the Extended Riemann Hypothesis, that the projection modulo p of any prefixed polynomial with integer coefficients can be completely factored in deterministic polynomial time if p − 1 has a (ln p) O(1)smooth divisor exceeding (p − 1) 1 2 +δ for some arbitrary small δ. We also address the issue of computing roots modulo p in deterministic time. 1