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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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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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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
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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Cited by 5 (1 self)
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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.
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.
Factoring Polynomials over Finite Fields using Balance Test
"... We study the problem of factoring univariate polynomials over finite fields. Under the assumption of the Extended Riemann Hypothesis (ERH), Gao [Gao01] designed a polynomial time algorithm that fails to factor only if the input polynomial satisfies a strong symmetry property, namely square balance. ..."
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We study the problem of factoring univariate polynomials over finite fields. Under the assumption of the Extended Riemann Hypothesis (ERH), Gao [Gao01] designed a polynomial time algorithm that fails to factor only if the input polynomial satisfies a strong symmetry property, namely square balance. In this paper, we propose an extension of Gao’s algorithm that fails only under an even stronger symmetry property. We also show that our property can be used to improve the time complexity of best deterministic algorithms on most input polynomials. The property also yields a new randomized polynomial time algorithm. 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 3 (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.
Deterministic polynomial factoring and association schemes. arXiv preprint arXiv:1205.5653
, 2012
"... Abstract. The problem of finding a nontrivial factor of a polynomial f(x) over a finite field Fq has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the generalized Riemann hypothesis (GRH). In this work we improv ..."
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Abstract. The problem of finding a nontrivial factor of a polynomial f(x) over a finite field Fq has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the generalized Riemann hypothesis (GRH). In this work we improve the state of the art by focusing on prime degree polynomials; let n be the degree. If (n − 1) has a ‘large ’ rsmooth divisor s, then we find a nontrivial factor of f(x) in deterministic poly(nr, log q) time; assuming GRH and that s = Ω( n/2r). Thus, for r = O(1) our algorithm is polynomial time. Further, for r = Ω(log logn) there are infinitely many prime degrees n for which our algorithm is applicable and better than the best known; assuming GRH. Our methods build on the algebraiccombinatorial framework of mschemes initiated by Ivanyos, Karpinski and Saxena (ISSAC 2009). We show that the mscheme on n points, implicitly appearing in our factoring algorithm, has an exceptional structure; leading us to the improved time complexity. Our
doi:10.1112/S1461157013000296 Deterministic
"... polynomial factoring and association schemes ..."
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