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Primitive polynomials, singer cycles and . . .
, 2011
"... Using the structure of Singer cycles in general linear groups, we prove that a conjecture of Zeng et al. (WordOriented Feedback Shift Register: σLFSR, 2007) holds in the affirmative in a special case, and outline a plausible approach to prove it in the general case. This conjecture is about the nu ..."
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the number of primitive σLFSRs of a given order over a finite field, and it generalizes a known formula for the number of primitive LFSRs, which, in turn, is the number of primitive polynomials of a given degree over a finite field. Moreover, this conjecture is intimately related to an open question
Study of Algorithms for Primitive Polynomials
, 1994
"... this report we shall present the fundamentals of random number generation on parallel processors. We shall exhibit how the practical task of carrying out stochastic simulation on a parallel machine leads deeply into number theory and algebra. We shall see that some classical algorithms which have pr ..."
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Cited by 2 (1 self)
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this report we shall present the fundamentals of random number generation on parallel processors. We shall exhibit how the practical task of carrying out stochastic simulation on a parallel machine leads deeply into number theory and algebra. We shall see that some classical algorithms which have proved to be excellent for singleprocessor machines, are either useless or require greatest care in the case of parallel processors. Stochastic simulation is one of the important tasks for single as well as multiprocessor machines. Computer simulations of reallife processes based on stochastic models have become one of the most interesting  and demanding  applications of mathematics. Due to the computational complexity of the problems, parallelization of the underlying algorithms is receiving increasing attention. As a basic condition to any research, we should be able to reproduce and to verify a scientific experiment. These two requirements and, further, considerations of storage and computational effectiveness rule out physical sources for random numbers, such as radioactive decay or electronic noise. The efficient generation of random numbers of high statistical quality is an absolute necessity for stochastic simulation. In his wellknown monograph, Ripley [19, p.2] writes: "The first thing needed for a stochastic simulation is a source of randomness. This is often taken for granted but is of fundamental importance. Regrettably many of the socalled random functions supplied with the most widespread computers are far from random, and many simulation studies have been invalidated as a consequence." D5H1/Rel 1.0/April 27, 1994 Random number generators for parallel processors PACT The following statement from Ripley[19, p.14] does not exaggerate the actual situation:...
Existence of primitive polynomials with three coefficients prescribed
 JP J. Algebra Number Theory Appl
"... Let Fq denote the finite field of q elements, q = pr for prime p and positive integer r. A monic polynomial f(x) = xn + ∑n i=1 fixn−i ∈ Fq[x] is called a primitive polynomial if it is irreducible over Fq and any of the roots of f can be used to ..."
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Cited by 2 (1 self)
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Let Fq denote the finite field of q elements, q = pr for prime p and positive integer r. A monic polynomial f(x) = xn + ∑n i=1 fixn−i ∈ Fq[x] is called a primitive polynomial if it is irreducible over Fq and any of the roots of f can be used to
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 961 (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
On the Construction of mSequences via Primitive Polynomials with a Fast Identification Method
"... Abstract—The paper provides an indepth tutorial of mathematical construction of maximal length sequences (msequences) via primitive polynomials and how to map the same when implemented in shift registers. It is equally important to check whether a polynomial is primitive or not so as to get proper ..."
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Abstract—The paper provides an indepth tutorial of mathematical construction of maximal length sequences (msequences) via primitive polynomials and how to map the same when implemented in shift registers. It is equally important to check whether a polynomial is primitive or not so as to get
Primitive Polynomials for Robust Scramblers and Stream Ciphers Against Reverse Engineering
"... Abstract—A linear feedback shift register (LFSR) is a basic component of a linear scrambler and a stream cipher for a communication system. And primitive polynomials are used as the feedback polynomials of the LFSRs. In a noncooperative context, the reverseengineering of a linear scrambler and a s ..."
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Abstract—A linear feedback shift register (LFSR) is a basic component of a linear scrambler and a stream cipher for a communication system. And primitive polynomials are used as the feedback polynomials of the LFSRs. In a noncooperative context, the reverseengineering of a linear scrambler and a
Results on multiples of primitive polynomials and their products over GF(2). Theoretical Computer Science 341(13
 GF (2). Theoretical Computer Science
, 2005
"... A standard model of nonlinear combiner generator for stream cipher system combines the outputs of several independent Linear Feedback Shift Register (LFSR) sequences using a nonlinear Boolean function to produce the key stream. Given such a model, cryptanalytic attacks have been proposed by finding ..."
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Cited by 3 (0 self)
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out the sparse multiples of the connection polynomials corresponding to the LFSRs. In this direction recently a few works are published on tnomial multiples of primitive polynomials. We here provide further results on degree distribution of the tnomial multiples. However, finding out the sparse
Primitive polynomials, singer cycles and wordoriented linear feedback shift registers
 DESIGNS CODES AND CRYPTOGRAPHY
, 2011
"... ..."
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