Interval Splitting Research Articles

Experimental Validation of an Ellipsoidal State Estimation Procedure for a Magnetic Levitation System

Set-based state estimation procedures have the advantage of enclosing all possible system states under the assumption of bounded measurement uncertainty, the structural correctness of dynamic systems models, and the representation of external disturbances and imperfectly known parameters by finitely large sets. In contrast to stochastic counterparts, often employing one of the available variants of Kalman filters, set-based approaches are less widely used. The reason for this observation is the fact that naive implementations often suffer from a non-negligible degree of overestimation and that (unless certain monotonicity properties are satisfied) set-based computations come with a notable increase of the computational complexity, resulting among others from required interval splitting procedures. This paper tries to resolve both issues by means of an ellipsoidal implementation of a discrete-time set-valued state estimation procedure that is validated experimentally and compared with an Unscented Kalman Filter (UKF) for a laboratory-scale magnetic levitation system.

A Taylor-Splitting Collocation approach and applications to linear and nonlinear engineering models

A novel matrix method based on the Taylor series called Taylor-Splitting Collocation Method is presented to solve linear and nonlinear ordinary differential equations. Unlike the previous approaches, the fundamental matrix equation is reformulated using interval splitting. The residual error estimation algorithm is presented. Convergence analysis is given in a general form. Four different mechanical models are analyzed:1.Forced oscillations of a linear spring-mass model2.Forced oscillations of a nonlinear spring-mass model3.Free oscillations of a cubic nonlinear spring-dashpot-mass model4.Forced oscillations of a damped nonlinear pendulum modelDisplacement-time and velocity-time dependencies are plotted for each model. Phase portraits of nonlinear models are presented. Appropriate absolute or residual error analyses are obtained for the proposed application models. The results of the new solution approach are compared with exact, numerical, and approximate solutions from previous works. Consistent results are found.

Energy Action Mechanism of Coal and Gas Outburst Induced by Rockburst

The essence of both rockburst and coal and gas outburst lies in fast energy release. In order to explore the energy action mechanism of coal and gas outburst induced by rockburst in rockburst and coal and gas outburst combined mines, the split Hopkinson pressure bar (SHPB) experimental device was firstly used to conduct uniaxial impact failure test of coal specimens prone to outburst under different strain rates, and their energy dissipation laws under impact loading were obtained. Next, under the engineering background of coal and gas dynamic phenomena induced by rockburst with different intensities in Xinyi Coal Mine and Pingdingshan Coal Group No. 12 Colliery in Henan Province and Dingji Coal Mine of Huainan Mining Group in Anhui Province, experimental study results were combined with numerical simulation analysis to discuss the energy mechanism of coal and gas outburst induced by rockburst. The study results show that the outburst can be divided into two different processes—critical outburst and outburst—according to the evolution law of outburst energy, and the critical energy conditions for coal and gas outburst are proposed. The minimum destructive energy range for the critical outburst of coal mass is obtained as (5–10) × 104 J/m3. Under some low gas, high stress, or strong disturbance conditions, applied loads can become the main energy sources causing critical failure and even crushing and throwing of coal mass. The coal mass will present an interval splitting structure under dynamic loading, which is obviously different from the failure mode of coal mass under static actions.

Approximation of Cauchy-type singular integrals with high frequency Fourier kernel

Two types of splitting algorithms are proposed for approximation of Cauchy type singular integrals having high frequency Fourier kernel. To evaluate non-singular integrals, modified Levin collocation methods with multiquadric radial basis function and Chebyshev polynomials are proposed. In the scenario of interval splitting, a multi-resolution quadrature is used to tackle the singularity ridden kernel. While in the case of integrand splitting, the singular part integral is evaluated analytically. Logarithmic singular integrals with oscillatory kernels are transformed to Cauchy principal value integrals and computed with the new algorithms. Error analysis of the component algorithms, as well as the individual methods, is performed theoretically. Validation of accuracy and error estimates of the methods are performed numerically as well.

ВЕЙВЛЕТ-ПРЕОБРАЗОВАНИЕ КАК МЕХАНИЗМ АНАЛИЗА СПЕКТРАЛЬНЫХ ХАРАКТЕРИСТИК КАРДИОИНТЕРВАЛОВ

Background. Today, when most processes and phenomena have a periodic structure that is not always homogeneous and depends on a number of irregular events, the classical spectral analysis based on the Fourier transform does not always allow an objective and reliable investigation. A promising alternative to the classical spectral analysis can be a wavelet transform, based on a well-localized function in both time and frequency domains, which allows us to analyze irregular processes. The purpose of the research was to analyze the potentials of the wavelet transform for the analysis of the spectral characteristics of cardiological time series with some heterogeneity; to develop a software application that allows performing wavelet analysis of data and building wavelet spectra, which are illustrative and easy in interpreting the obtained results. Material and methods. 37 cardiological time series were studied. We developed an application for the application package Mathematika 5.0 that allows building and analyzing the wavelet spectra of the studied data. Results. An alternative method for analyzing the spectral characteristics of the studied cardiological time series has been developed and clearly demonstrated. It allows observing the dynamics of the change in periods over the entire observation interval without performing the interval splitting, as would be required by the classical Fourier analysis. Conclusion. For the studied groups of patients the identified periods, found with the help of the wavelet transform have been presented. The advantages of using wavelets over the classical spectral analysis, which in cases of unstable time series does not allow tracking the dynamics of the changing in the data spectrum, have been demonstrated.

Table Substitution Box Method for Increasing Security in Interval Splitting Arithmetic Coding

Amalgamation of compression and security is indispensable in the field of multimedia applications. A novel approach to enhance security with compression is discussed in this  research paper. In secure arithmetic coder (SAC), security is provided by input and output permutation methods and compression is done by interval splitting arithmetic coding. Permutation in SAC is susceptible to attacks. Encryption issues associated with SAC is dealt in this research method. The aim of this proposed method is to encrypt the data first by Table Substitution Box (T-box) and then to compress by Interval Splitting Arithmetic Coder (ISAC). This method incorporates dynamic T-box in order to provide better security. T-box is a method, constituting elements based on the random output of Pseudo Random Generator (PRNG), which gets the input from Secure Hash Algorithm-256 (SHA-256) message digest. The current scheme is created, based on the key, which is known to the encoder and decoder. Further, T-boxes are created by using the previous message digest as a key.  Existing interval splitting arithmetic coding of SAC is applied for compression of text data. Interval splitting finds a relative position to split the intervals and this in turn brings out compression. The result divulges that permutation replaced by T-box method provides enhanced security than SAC. Data is not revealed when permutation is replaced by T-box method. Security exploration reveals that the data remains secure to cipher text attacks, known plain text attacks and chosen plain text attacks. This approach results in increased security to Interval ISAC. Additionally the compression ratio is compared by transferring the outcome of T-box to traditional arithmetic coding. The comparison proved that there is a minor reduction in compression ratio in ISAC than arithmetic coding. However the security provided by ISAC overcomes the issues of compression ratio in arithmetic coding.Â

Choices and intervals

We consider a random interval splitting process, in which the splitting rule depends on the empirical distribution of interval lengths. We show that this empirical distribution converges to a limit almost surely as the number of intervals goes to infinity. We give a characterization of this limit as a solution of an ODE and use this to derive precise tail estimates. The convergence is established by showing that the size-biased empirical distribution evolves in the limit according to a certain deterministic evolution equation. Although this equation involves a non-local, non-linear operator, it can be studied thanks to a carefully chosen norm with respect to which this operator is contractive. In finite-dimensional settings, convergence results like this usually go under the name of stochastic approximation and can be approached by a general method of Kushner and Clark. An important technical contribution of this article is the extension of this method to an infinite-dimensional setting.

Perturbation meets key‐based interval splitting arithmetic coding: security enhancement and chaos generalization

AbstractKey‐based interval splitting arithmetic coding (KAC) possesses both encryption and compression capabilities. However, it possesses vulnerability to chosen‐plaintext attack because the attacker can explore the relationship between the key and the codeword to deduce the secret key. In order to resist this attack, we propose to introduce perturbation into KAC. The perturbation‐based KAC not only avoids the flaw of KAC that the splitting keys are usually located at the endpoint of certain codeword or at the border of two codewords but also removes the restriction that the keys are only allowed in certain sub‐intervals, which result in great convenience to the key scheduler. In addition, based on generalized arithmetic coding using Generalized Luröth Series, we study the phase‐space splitting of a chaotic map for generalized KAC and suggest the generalized perturbation‐based KAC. This leads to the design of a joint compression and encryption scheme with more powerful cryptographic features. Copyright © 2015 John Wiley & Sons, Ltd.

Computing Cumulative Rewards Using Fast Adaptive Uniformization

The computation of transient probabilities for continuous-time Markov chains often employs uniformization, also known as the Jensen method. The fast adaptive uniformization method introduced by Mateescu et al. approximates the probability by neglecting insignificant states and has proven to be effective for quantitative analysis of stochastic models arising in chemical and biological applications. However, this method has only been formulated for the analysis of properties at a given point of time t . In this article, we extend fast adaptive uniformization to handle expected reward properties that reason about the model behavior until time t , for example, the expected number of chemical reactions that have occurred until t . To show the feasibility of the approach, we integrate the method into the probabilistic model checker PRISM and apply it to a range of biological models. The performance of the method is enhanced by the use of interval splitting. We compare our implementation to standard uniformization implemented in PRISM and to fast adaptive uniformization without support for cumulative rewards implemented in MARCIE, demonstrating superior performance.

An image joint compression-encryption algorithm based on adaptive arithmetic coding

Through a series of studies on arithmetic coding and arithmetic encryption, a novel image joint compression-encryption algorithm based on adaptive arithmetic coding is proposed. The contexts produced in the process of image compression are modified by keys in order to achieve image joint compression encryption. Combined with the bit-plane coding technique, the discrete wavelet transform coefficients in different resolutions can be encrypted respectively with different keys, so that the resolution selective encryption is realized to meet different application needs. Zero-tree coding is improved, and adaptive arithmetic coding is introduced. Then, the proposed joint compression-encryption algorithm is simulated. The simulation results show that as long as the parameters are selected appropriately, the compression efficiency of proposed image joint compression-encryption algorithm is basically identical to that of the original image compression algorithm, and the security of the proposed algorithm is better than the joint encryption algorithm based on interval splitting.

Transmission protocol selection for dual‐hop amplify‐and‐forward relaying networks

This study solves the problem of transmission protocol selection for dual-hop amplify-and-forward relaying networks. The two available protocols are parallel and non-parallel, and the selection metric is the system data rate. First, for the non-parallel protocol, the joint optimal power allocation (PA) scheme is derived using a single parameter optimisation approach. Secondly, for the parallel protocol, the optimal PA scheme is obtained through introducing a novel interval splitting approach, which considerably reduces the solution space from exponential to polynomial order. Finally, based on the proposed optimal PA schemes, the rate performance of the two protocols are compared, which reveals that their relationship is decided by the given average power constraint. The critical value of the average power constraint, above which the performance of the non-parallel protocol outperforms that of the parallel protocol, is determined via numerical simulations. Therefore the critical value provides a selection criterion between the two protocols under data rate metric.

On the uniform distribution modulo 1 of multidimensional LS-sequences

Ingrid Carbone introduced the notion of so-called LS-sequences of points, which are obtained by a generalization of Kakutani’s interval splitting procedure. Under an appropriate choice of the parameters \(L\) and \(S\), such sequences have low discrepancy, which means that they are natural candidates for Quasi-Monte Carlo integration. It is tempting to assume that LS-sequences can be combined coordinatewise to obtain a multidimensional low-discrepancy sequence. However, in the present paper, we prove that this is not always the case: if the parameters \(L_1,S_1\) and \(L_2,S_2\) of two one-dimensional low-discrepancy LS-sequences satisfy certain number-theoretic conditions, then their two-dimensional combination is not even dense in \([0,1]^2\).

On the Security of Key-Based Interval Splitting Arithmetic Coding With Respect to Message Indistinguishability

Key-based interval splitting arithmetic coding (KSAC) has been proposed to improve the security of traditional arithmetic coding (AC). Chosen-plaintext attacks have been proposed for KSAC when the same key is used to encrypt different messages. In this paper, we consider a stronger version of KSAC, where different keys are used to encrypt different messages. We then use message indistinguishability to prove that this version of KSAC is insecure under ciphertext-only attacks, a weaker form of attack than chosen-plaintext attacks. Indistinguishability in the presence of an eavesdropper is a security definition equivalent to semantic security. We prove the insecurity over the alphabet {A,B} with pB=(1/2(1+2s)) and pA=1-pB where pA and pB are the probabilities of the source generating A and B, respectively, and s is the number of bits in each splitting key.

Gaussian integration with rescaling of abscissas and weights

An algorithm for integration of the polynomial functions with a variable weight is considered. It provides an extension of the Gaussian integration, with appropriate scaling of the abscissas and weights. In a first step, orthogonal polynomials are computed for a fixed a = 1 . Then, using approximate scaling, the initial guess is constructed for a ≠ 1 . Finally, numerical values of the abscissas and weights are refined, solving polynomial system using Newton–Raphson method. The final form of the algorithm provides good alternative to usually adopted interval splitting, automatically avoiding problems with limiting values of parameter present in the weight function. Construction of the method requires arbitrary precision arithmetic and special functions, polylogarithms in particular. The final form of the algorithm can be coded using machine precision floating point numbers and standard mathematical library.

On the Security of Randomized Arithmetic Codes Against Ciphertext-Only Attacks

Modifications of arithmetic coding (AC) have been proposed to improve the security of traditional AC. Two main modifications to AC are randomized AC (RAC) and AC with key-based interval splitting (KSAC). Chosen-plaintext attacks have been proposed for these two methods when the same key is used to encrypt different messages. We first give a definition for security of encryption using AC that is based on the inability of the adversary to distinguish between the encryption of one plaintext from the encryption of another. Using this definition, we prove that RAC is insecure even if a new random key is used to compress every message. Our proof assumes that the adversary can only eavesdrop on the ciphertext and cannot request encryptions of chosen-plaintexts. We then prove that the method of first-compress-then-encrypt, where the encryption is performed by a bitwise xor of the compressed output with a pseudorandom bit sequence, is provably secure with respect to chosen-plaintext attacks. If the pseudorandom bit sequence is derived in advance using Advanced Encryption Standard (AES) in the counter mode, then the first-compress-then-encrypt method results in a performance penalty of only a few two input xor-gate delays.

Adaptive Chosen-Ciphertext Attack on Secure Arithmetic Coding

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> The paper “Secure Arithmetic Coding” (in <emphasis emphasistype="bold"><emphasis emphasistype="smcaps">IEEE Transactions on Signal Processing</emphasis></emphasis>, vol. 55, no. 5, pp. 2263–2272, May 2007) presented a novel encryption scheme called the secure arithmetic coding (SAC) based on the interval splitting arithmetic coding (ISAC) and a series of permutations. In the current work, we study the security of the SAC under an adaptive chosen-ciphertext attack. It is shown that the key vectors used in the codeword permutation step can be recovered with complexity <formula formulatype="inline"><tex Notation="TeX">${\cal O}(N)$</tex></formula>, where <formula formulatype="inline"><tex Notation="TeX">$N$</tex> </formula> is the symbol sequence length. After getting these key vectors, we can remove the codeword permutation step, and the resulting system has already been shown to be insecure in the original paper. This implies that the SAC is not suitable for the applications where the attacker can have access to the decoder. In addition, we discuss a method to jointly enhance the security and the performance of the SAC. </para>

Verified Real Number Calculations: A Library for Interval Arithmetic

Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly automated as well as interactive way. First, we formally establish upper and lower bounds for elementary functions. Then, based on these bounds, we develop a rational interval arithmetic where real number calculations take place in an algebraic setting. In order to reduce the dependency effect of interval arithmetic, we integrate two techniques: interval splitting and Taylor series expansions. This pragmatic approach has been developed, and formally verified, in a theorem prover. The formal development also includes a set of customizable strategies to automate proofs involving explicit calculations over real numbers. Our ultimate goal is to provide guaranteed proofs of numerical properties with minimal human theorem-prover interaction.

Cryptanalysis of Some Multimedia Encryption Schemes

Encryption is one of the fundamental technologies that is used in digital rights management. Unlike ordinary computer applications, multimedia applications generate large amounts of data that has to be processed in real time. So, a number of encryption schemes for multimedia applications have been proposed in recent years. We analyze the following proposed methods for multimedia encryption: key-based multiple Huffman tables (MHT), arithmetic coding with key-based interval splitting (KSAC), and randomized arithmetic coding (RAC). Our analysis shows that MHT and KSAC are vulnerable to low complexity known- and/or chosen-plaintext attacks. Although we do not provide any attacks on RAC, we point out some disadvantages of RAC over the classical compress-then-encrypt approach.

Binary arithmetic coding with key-based interval splitting

Binary arithmetic coding involves recursive partitioning the range [0,1) in accordance with the relative probabilities of occurrence of the two input symbols. We describe a modification of this approach in which the overall length within the range [0,1) allocated to each symbol is preserved, but the traditional assumption that a single contiguous interval is used for each symbol is removed. A key known to both the encoder and decoder is used to describe where the intervals are "split" prior to encoding each new symbol. The repeated splitting has the effect of both scrambling the intervals and altering their lengths, thereby allowing both encryption and compression to be obtained simultaneously.

Weak convergence results for the Kakutani interval splitting procedure

This paper obtains the weak convergence of the empirical processes of both the division points and the spacings that result from the Kakutani interval splitting model. In both cases, the limit processes are Gaussian. For the division points themselves, the empirical processes converge to a Brownian bridge as they do for the usual uniform splitting model, but with the striking difference that its standard deviations are about one-half as large. This result gives a clear measure of the degree of greater uniformity produced by the Kakutani model. The limit of the empirical process of the normalized spacings is more complex, but its covariance function is explicitly determined. The method of attack for both problems is to obtain first the analogous results for more tractable continuous parameter processes that are related through random time changes. A key tool in their analysis is an approximate Poissonian characterization that obtains for cumulants of a family of random variables that satisfy a specific functional equation central to the K-model.