The k-Error Linear Complexity and the Linear Complexity for pq^n-Periodic Binary Sequences

The k-error linear complexity and the linear complexity of the keystream of a stream cipher are two important standards to scale the randomness of the key stream. For a pq^n-periodic binary sequences where p, q are two odd primes satisfying that 2 is a primitive root module p and q^2 and gcd（p-1, q-1） = 2, we analyze the relationship between the linear complexity and the minimum value k for which the k-error linear complexity is strictly less than the linear complexity.

On k-Error Linear Complexity of Some Explicit Nonlinear Pseudorandom Sequences

Combining with the research on the linear complexity of explicit nonlinear generators of pseudorandom sequences, we study the stability on linear complexity of two classes of explicit inversive generators and two classes of explicit nonlinear generators. We present some lower bounds in theory on the k-error linear complexity of these explicit generatol＇s, which further improve the cryptographic properties of the corresponding number generators and provide very useful information when they are applied to cryptography.

THE 2-ERROR LINEAR COMPLEXITY OF 2~n-PERIODIC BINARY SEQUENCES WITH LINEAR COMPLEXITY 2~n-1

Linear complexity and k-error linear complexity of the stream cipher are two important standards to scale the randomicity of keystreams. For the 2n -periodicperiodic binary sequence with linear complexity 2n 1and k = 2,3,the number of sequences with given k-error linear complexity and the expected k-error linear complexity are provided. Moreover,the proportion of the sequences whose k-error linear complexity is bigger than the expected value is analyzed.

Linear Complexity of Some Binary Sequences Derived from Fermat Quotients

We determined the linear complexity of a family of p2-periodic binary threshold sequences and a family of p2-periodic binary sequences constructed using the Legendre symbol,both of which are derived from Fermat quotients modulo an odd prime p.If 2 is a primitive element modulo p2,the linear complexity equals to p2-p or p2-1,which is very close to the period and it is large enough for cryptographic purpose.

ON THE LINEAR COMPLEXITY OF FCSR SEQUENCES

The linear complexity of a new kind of keystream sequences.FCSR sequences,is discussed by use of the properties of cyclotomic polynomials.Based on the results of C.Seo's,an upper bound and a lower bound on the linear complexity of a significant kind of FCSR sequences—l-sequences are presented.

Linear complexity of Ding generalized cyclotomic sequences

Minimal polynomials and linear complexity of binary Ding generalized cyclotomic sequences of order 2 with the two-prime residue ring Zpq are obtained by Bai in 2005. In this paper, we obtain linear complexity and minimal polynomials of all Ding generalized cyclotomic sequences. Our result shows that linear complexity of these sequences takes on the values pq and pq-1 on our necessary and sufficient condition with probability 1/4 and the lower bound （pq - 1）/2 with probability 1/8. This shows that most of these sequences are good. We also obtained that linear complexity and minimal polynomials of these sequences are independent of their orders. This makes it no more difficult in choosing proper p and q.

Linear complexity and autocorrelation of a new class of binary generalized cyclotomic sequences of order two and length pqr

Cyclotomic sequences have good cryptographic properties and are closely related to difference sets.This paper proposes a new class of binary generalized cyclotomic sequences of order two and length pqr.Its linear complexity,minimal polynomial,and autocorrelation are investigated.The results show that these sequences have a large linear complexity when 2∈D1,which means they can resist the Berlekamp-Massey attack.Furthermore,the autocorrelation values are close to 0 with a probability of approximately 1?1/r.Therefore,when r is a big prime,the new sequence has a good autocorrelation.

Scale, Complexity, and Cybersecurity Risk Management

Elementary information theory is used to model cybersecurity complexity, where the model assumes that security risk management is a binomial stochastic process. Complexity is shown to increase exponentially with the number of vulnerabilities in combination with security risk management entropy. However, vulnerabilities can be either local or non-local, where the former is confined to networked elements and the latter results from interactions between elements. Furthermore, interactions involve multiple methods of communication, where each method can contain vulnerabilities specific to that method. Importantly, the number of possible interactions scales quadratically with the number of elements in standard network topologies. Minimizing these interactions can significantly reduce the number of vulnerabilities and the accompanying complexity. Two network configurations that yield sub-quadratic and linear scaling relations are presented.

THE APPLICATION OF LINEAR COMPLEXITY OF SEQUENCES TO LOWER BOUNDS ON THE MINIMUM DISTANCE OF CYCLIC CODES

Firstly,the Fourier transforms in finite fields and the concept of linear complexityof sequences are described.Then several known lower bounds on the minimum distance of cycliccodes are outlined.Finally,the minimum distance of cyclic codes is analyzed via linear complexityof sequences,and new theorems about the lower bounds are obtained.

Linear Complexity of New Generalized Cyclotomic Sequences of Length 2pq

The linear complexity and minimal polynomial of new generalized cyclotomic sequences of order two are investigated.A new generalized cyclotomic sequence Sof length 2pqis defined with an imbalance p+1.The results show that this sequence has high linear complexity.

Quasi-optimal complexity of adaptive finite element method for linear elasticity problems in two dimensions

This paper introduces an adaptive finite element method （AFEM） using the newest vertex bisection and marking exclusively according to the error estimator without special treatment of oscillation. By the combination of the global lower bound and the localized upper bound of the posteriori error estimator, perturbation of oscillation, and cardinality of the marked element set, it is proved that the AFEM is quasi-optimal for linear elasticity problems in two dimensions, and this conclusion is verified by the numerical examples.

THE DISTRIBUTION FUNCTIONS FOR THE LINEAR COMPLEXITY OF PERIODIC SEQUENCES

Linear complexity is an important standard to scale the randomicity of stream ciphers. The distribution function of a sequence complexity measure gives the function expression for the number of sequences with a given complexity measure value. In this paper, we mainly determine the distribution function of sequences with period over using Discrete Fourier Transform (DFT), where and the characteristics of are odd primes, gcd and is a primitive root modulo The results presented can be used to study the randomness of periodic sequences and the analysis and design of stream cipher.

Polynomial Complexity Bounds of Mehrotra-type Predictor-corrector Algorithms for Linear Programming over Symmetric Cones

We establish polynomial complexity corrector algorithms for linear programming over bounds of the Mehrotra-type predictor- symmetric cones. We first slightly modify the maximum step size in the predictor step of the safeguard based Mehrotra-type algorithm for linear programming, that was proposed by Salahi et al. Then, using the machinery of Euclidean Jordan algebras, we extend the modified algorithm to symmetric cones. Based on the Nesterov-Todd direction, we obtain O（r log ε1） iteration complexity bound of this algorithm, where r is the rank of the Jordan algebras and ε is the required precision. We also present a new variant of Mehrotra-type algorithm using a new adaptive updating scheme of centering parameter and show that this algorithm enjoys the same order of complexity bound as the safeguard algorithm. We illustrate the numerical behaviour of the methods on some small examples.

ZEROS OF ENTIRE SOLUTIONS TO COMPLEX LINEAR DIFFERENCE EQUATIONS

In this article, we study the complex oscillation problems of entire solutions to homogeneous and nonhomogeneous linear difference equations, and obtain some relations of the exponent of convergence of zeros and the order of growth of entire solutions to complex linear difference equations.

SYNCHRONIZATION OF MASTER-SLAVE MARKOVIAN SWITCHING COMPLEX DYNAMICAL NETWORKS WITH TIME-VARYING DELAYS IN NONLINEAR FUNCTION VIA SLIDING MODE CONTROL

In this article, a synchronization problem for master-slave Markovian switching complex dynamical networks with time-varying delays in nonlinear function via sliding mode control is investigated. On the basis of the appropriate Lyapunov-Krasovskii functional, introducing some free weighting matrices, new synchronization criteria are derived in terms of linear matrix inequalities （LMIs）. Then, an integral sliding surface is designed to guarantee synchronization of master-slave Markovian switching complex dynamical networks, and the suitable controller is synthesized to ensure that the trajectory of the closed-loop error system can be driven onto the prescribed sliding mode surface. By using Dynkin＇s formula, we established the stochastic stablity of master-slave system. Finally, numerical example is provided to demonstrate the effectiveness of the obtained theoretical results.

PREPARATION AND CHARACTERIZATION OF INCLUSION COMPLEXES OF TWO-ARM LINEAR AND FOUR-ARM STAR-SHAPED POLY(ε-CAPROLACTONE)S WITHα-CYCLODEXTRIN

Both four-ann star-shaped poly（ε-caprolactone） （4sPCL） and two-ann linear PCL （2LPCL） were synthesized and their inclusion complexation with α-cyclodextrin （α-CD） were studied. The inclusion complexes （ICs） formed between the PCL polymers and α-CD were characterized by ^1H-NMR, DSC, TGA, WAXD, and FT-1R, respectively. Both branch ann number and molecular weight of the PCL polymers have apparent effect on the stoichiometry （CL：CD, mol：mol） of these ICs. All these analytical results indicate that the branch arms of the PCL polymers are incorporated into the hydrophobic α-CD cavities and their original crystalline properties are completely suppressed. Moreover, the inclusion complexation between two-ann linear or four-ann star-shaped PCL polymers and α-CD not only enhances the thermal stability of the guest PCL polymers but also improves that of α-CD.

Statistical Characteristics of the Complexity for Prime-Periodic Sequences

Using the fact that the factorization of x^N — 1 over GF(2) is especiallyexplicit, we completely establish the distributions and the expected values of the lineal complexityand the k-error linear complexity of the N-periodic sequences respectively,where N is an odd primeand 2 is a primitive root modulo N. The results show that there are a large percentage of sequenceswith both the linear complexity and the k-enor linear complexity not less than N, quite close totheir maximum possible values.

STUDY OF REGULARITY OF GENERAL-LINEAR-COMPLEX SINGULARITY OF 3/6-STEWART MECHANISM

The general-linear-complex singularity of Stewart mechanism is a veryimportant problem in the parallel manipulator. Its general regularity is not found yet during thepast two decades. St-Onge and Gosselin pointed Out that the singularity locus of the Stewartmechanism at some given orientations of the moving platform should be polynomial expressions withvaried degrees in 2000, but they didn't formulate the expression. Based on the kinematicssingularity principle and the geometry condition proposed by Huang Zhen in 1999, firstly thesingularity equation in degree two is derived. It is a hyperbola when the orientation of the movingplatform is given. This result is also proved using screw theory. Then some singularity surfaces aregotten in three-dimensional space. This result is of important significance.

Short communication Linear analysis on the stability of lanthanide vapor complexes LnAl_3Cl_(12)(Ln=LatoLu)

The reactions LnCl3 (s) + (3/2)Al2Cl6, (g) = LnAl3Cl12 (g) for Ln = La to Lu were studied by quenching experiments in roughly the same temperature and pressure ranges (588-851 K and 0.01-0.22 MPa). Stability constants Kθ of lanthanide complexes LnAl3Cl12 were calculated from the measurements. The values of Ig Kθ change linearly with the ionpotential (Z+/r) of lanthanide(Ⅲ) from La to Gd and from Tb to Lu, respectively, indicating the Gd break. There exist inclined W effect between Ig Kθ and the total angular momentum L of lanthanide(Ⅲ). And hereby lanthanide elements are divided into four segments, La-Nd, Pm-Gd, Tb-Ho, and Er-Lu. In each segment, the linearity is maintained.

Synchronization of High-order Discrete-time Linear Complex Networks with Time-varying Delays

Synchronization of high-order discrete-time complex networks with undirected topologies is studied and the impacts of time delays are investigated. Firstly,by the state decomposition,synchronization problems are transformed into asymptotic stability ones of multiple lower dimensional time-delayed subsystems. Then,linear matrix inequality( LMI) criteria for synchronization are given,which can guarantee the scalability of complex networks since they only include three LMI constraints independent of the number of agents. Moreover,an explicit expression of the synchronization function is presented,which can describe the synchronization behavior of all agents in complex networks. Finally,a numerical example is given to demonstrate the theoretical results,where it is shown that if the gain matrices of synchronization protocols satisfy LMI criteria for synchronization,synchronization can be achieved.