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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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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 ...
Improved Incremental Prime Number Sieves
 Cornell University
, 1994
"... . An algorithm due to Bengalloun that continuously enumerates the primes is adapted to give the first prime number sieve that is simultaneously sublinear, additive, and smoothly incremental:  it employs only \Theta(n= log log n) additions of numbers of size O(n) to enumerate the primes up to n, ..."
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. An algorithm due to Bengalloun that continuously enumerates the primes is adapted to give the first prime number sieve that is simultaneously sublinear, additive, and smoothly incremental:  it employs only \Theta(n= log log n) additions of numbers of size O(n) to enumerate the primes up to n, equalling the performance of the fastest known algorithms for fixed n;  the transition from n to n + 1 takes only O(1) additions of numbers of size O(n). (On average, of course, O(1) such additions increase the limit up to which all primes are known from n to n + \Theta(log log n)). 1 Introduction A socalled "formula" for the i'th prime has been a longlived concern, if not quite the Holy Grail, of Elementary Number Theory. This concern seems poorly motivated, as evidenced by the extraordinary freakshow of solutions proffered over the ages. The natural setting is Algorithmic Number Theory, and what is desired is much better cast as an algorithm to compute the i'th prime. Given that app...
Fast construction of irreducible polynomials over finite fields (version of 22 Apr
, 2009
"... We present a randomized algorithm that on input a finite field K with q elements and a positive integer d outputs a degree d irreducible polynomial in K[x]. The running time is d 1+o(1) × (log q) 5+o(1) elementary operations. The o(1) in d 1+o(1) is a function of d that tends to zero when d tends to ..."
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Cited by 3 (0 self)
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We present a randomized algorithm that on input a finite field K with q elements and a positive integer d outputs a degree d irreducible polynomial in K[x]. The running time is d 1+o(1) × (log q) 5+o(1) elementary operations. The o(1) in d 1+o(1) is a function of d that tends to zero when d tends to infinity. And the o(1) in (log q) 5+o(1) is a function of q that tends to zero when q tends to infinity. In particular, the complexity is quasilinear in the degree d. 1
The Modulo N Extended GCD Problem for Polynomials
 IN ISSAC'98
, 1997
"... We study the following problem: Given a; b; N 2 F [x] with gcd(a; b; N) = 1 and N nonzero, compute a minimal degree f 2 F [x] which satisfies gcd(a + fb; N) = 1. We give a deterministic algorithm for solving this problem that is applicable over any field. The algorithm is designed to solve efficien ..."
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Cited by 2 (0 self)
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We study the following problem: Given a; b; N 2 F [x] with gcd(a; b; N) = 1 and N nonzero, compute a minimal degree f 2 F [x] which satisfies gcd(a + fb; N) = 1. We give a deterministic algorithm for solving this problem that is applicable over any field. The algorithm is designed to solve efficiently a succession of such problems for a fixed N . When q = #F ? deg N the solution will satisfy deg f = 0. When q deg N we conjecture that the solution satisfies deg f dlog q deg Ne; in this case the complexity bound we give for the algorithm depends on this conjecture. As an application we demonstrate a deterministic algorithm for computing transforming matrices for the Smith normal form of a nonsingular A 2 F [x] n\Thetan . When q is too small most previous algorithms require working over an algebraic extension of F and may not produce transforming matrices over F [x]. The algorithm we propose will produce transforming matrices over F [x], for fields F of any size.
Algorithms for large integer matrix problems. In Applied algebra, algebraic algorithms and errorcorrecting codes
 of Lecture
"... Abstract. New algorithms are described and analysed for solving various problems associated with a large integer matrix: computing the Hermite form, computing a kernel basis, and solving a system of linear diophantine equations. The algorithms are spaceefficient and for certain types of input matri ..."
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Abstract. New algorithms are described and analysed for solving various problems associated with a large integer matrix: computing the Hermite form, computing a kernel basis, and solving a system of linear diophantine equations. The algorithms are spaceefficient and for certain types of input matrices — for example, those arising during the computation of class groups and regulators — are faster than previous methods. Experiments with a prototype implementation support the running time analyses. 1
Modular fibers and illumination problems
, 2005
"... Abstract. For a Veech surface (X, ω), we characterize Aff + (X, ω) invariant subspaces of X n and prove that nonarithmetic Veech surfaces have only finitely many invariant subspaces of very particular shape (in any dimension). Among other consequences we find copies of (X, ω) embedded in the moduli ..."
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Abstract. For a Veech surface (X, ω), we characterize Aff + (X, ω) invariant subspaces of X n and prove that nonarithmetic Veech surfaces have only finitely many invariant subspaces of very particular shape (in any dimension). Among other consequences we find copies of (X, ω) embedded in the modulispace of translation surfaces. We study illumination problems in (pre)lattice surfaces. For (X, ω) prelattice we prove the at most countableness of points nonilluminable from any x ∈ X. Applying our results on invariant subspaces we prove the finiteness of these sets when (X, ω) is Veech. 1.
Elliptic periods for finite fields ∗
, 2008
"... We construct two new families of basis for finite field extensions. Bases in the first family, the socalled elliptic bases, are not quite normal bases, but they allow very fast Frobenius exponentiation while preserving sparse multiplication formulas. Bases in the second family, the socalled normal ..."
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We construct two new families of basis for finite field extensions. Bases in the first family, the socalled elliptic bases, are not quite normal bases, but they allow very fast Frobenius exponentiation while preserving sparse multiplication formulas. Bases in the second family, the socalled normal elliptic bases are normal bases and allow fast (quasilinear) arithmetic. We prove that all extensions admit models of this kind. 1
unknown title
, 2007
"... Jacobsthal’s function and a short proof of the density of a set in the unit hypercube ..."
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Jacobsthal’s function and a short proof of the density of a set in the unit hypercube