Composition Theorem Research Articles

Formal analysis of signal protocol based on logic of events theory

The Signal is an end-to-end encrypted communication protocol composed of a double ratchet (DR) protocol and an extended triple Diffie-Hellman (X3DH) protocol. Its complex ratchet structure and the characteristics of protocol composition make it challenging to realize formal analysis. A formal analysis method based on logic of events theory (LoET) is proposed to conduct a security analysis of the Signal protocol. The method includes inference rules with key relation and key chain as the core to realize the formal analysis of ratchet structure, and the inference relation between sub-protocols is established by putting forward the composition theorem. The proposed method achieves a formal analysis of Signal, revealing that it does not satisfy a strong authentication property during the X3DH phase. The results show that the LoET-based method can be effectively applied in the formal analysis of Signal protocols, thus promoting the application and development of these protocols with ratchet structure and composition properties.

Strong Asymptotic Composition Theorems for Mutual Information Measures

We characterize the growth of the Sibson and Arimoto mutual informations and α-maximal leakage, of any order that is at least unity, between a random variable and a growing set of noisy, conditionally independent and identically-distributed observations of the random variable. Each of these measures increases exponentially fast to a limit that is order- and measure-dependent, with an exponent that is order- and measure-independent.

On the Coincidence Theorem

We are proving Coincidence theorem due to Walsh for the case when the total degree of a polynomial is less than the number of arguments. Also, the following result has been proven: if $$p(z)$$ is a complex polynomial of degree $$n$$, then closed disk D that contains at least $$n-1$$ of its zeros (counting multiplicity) contains at least $$\left[\frac{n-2k+1}{2} \right]$$ zeros of its $$k$$-th derivative, provided that the arithmetical mean of these zeros is also centre of D. We also prove a variation of the classical composition theorem due to Szegö.

Square-mean pseudo almost automorphic solutions of infinite class under the light of measure theory

The aim of this work is to present new concept of square-mean pseudo almost automorphic of infinite class using the measure theory. We use the (µ,ν)-ergodic process to define the spaces of (µ,ν)-pseudo almost automorphic processes of infinite class in the square-mean sense. We present many interesting results on those spaces like completeness and composition theorems and we study the existence and the uniqueness of the square-mean (µ,ν)-pseudo almost automorphic solutions of infinite class for of the stochastic evolution equation. We provide an example to illustrate ours results.

WEIGHTED \(S^p\)-PSEUDO \(S\)-ASYMPTOTICALLY PERIODIC SOLUTIONS FOR SOME SYSTEMS OF NONLINEAR DELAY INTEGRAL EQUATIONS WITH SUPERLINEAR PERTURBATION

This work is concerned with the existence of positive weighted pseudo \(S\)-asymptotically periodic solution in Stepanov-like sense for some systems of nonlinear delay integral equations. In this context, we will first be interested in establishing a suitable composition theorem, and then some existing results concerning the \(S\)-asymptotic periodicity in the scalar case are developed here for the vector case. We point out that, in this paper, we adopt some changes in the definitions, which, although slight, are necessary to accomplish the work.

The measure pseudo almost-periodicty and automorphy in HOHNNs with time-varying delays

In this work, we prove some existence, global exponential stability and uniqueness results for measure pseudo almost periodic and automorphic solutions to a class of high-order Hopfield neural networks with delays. The main technique is based upon some appropriate composition theorems combined with the Banach contraction mapping principle. Finally, three numerical examples are given to illustrate the effectiveness of our results.

Noiseless Privacy: Definition, Guarantees, and Applications

In this article, we define noiseless privacy, as a non-stochastic rival to differential privacy, requiring that the outputs of a mechanism (i.e., function composition of a privacy-preserving mapping and a query) attain only a few values while varying the data of an individual (the logarithm of the number of the distinct values is bounded by the privacy budget). Therefore, the output of the mechanism is not fully informative of the data of the individuals in the dataset. We prove several guarantees for noiselessly-private mechanisms. The information content of the output about the data of an individual, even if an adversary knows all the other entries of the private dataset, is bounded by the privacy budget. The zero-error capacity of memory-less channels using noiselessly private mechanisms for transmission is upper bounded by the privacy budget. The performance of a non-stochastic hypothesis-testing adversary is bounded again by the privacy budget. Assuming that an adversary has access to a stochastic prior on the dataset, we prove that the estimation error of the adversary for individual entries of the dataset is lower bounded by a decreasing function of the privacy budget. In this case, we also show that the maximal leakage is bounded by the privacy budget. In addition to privacy guarantees, we prove that noiselessly-private mechanisms admit composition theorem and post-processing does not weaken their privacy guarantees. We prove that quantization or binning can ensure noiseless privacy if the number of quantization levels is appropriately selected based on the sensitivity of the query and the privacy budget. Finally, we illustrate the privacy merits of noiseless privacy using multiple datasets in energy, transport, and finance.

New composition theorem for weighted Stepanov-like pseudo almost periodic functions on time scales and applications

First, we show a new composition theorem for both Stepanov almost periodic functions and for weighted Stepanov-like pseudo almost periodic functions on time scales. Next, under some suitable assumptions, we study the existence and uniqueness of weighted pseudo almost periodic solutions to some first-order dynamic equations on time scales with weighted Stepanov-like pseudo almost periodic coefficients.

The doubly weighted pseudo‐almost automorphic mild solutions to a class of stochastic dynamic equations

This paper first obtains the necessary and sufficient condition between Stepanov‐like doubly weighted pseudo‐almost automorphic stochastic processes and Stepanov‐like asymptotically almost automorphic stochastic processes in a suitable set and, second, establishes the composition theorem based on ‐measurability for nonequivalent weight functions; these new conclusions acquired perfect and extended some pre‐existing related results, which enrich the dynamics of Stepanov‐like doubly weighted pseudo‐almost automorphic stochastic processes. Finally, for a class of nonautonomous stochastic equations with the coefficients of Stepanov‐like doubly weighted pseudo‐almost automorphic stochastic processes that satisfies non‐Lipschitz conditions, we investigate the existence and uniqueness of the doubly weighted pseudo‐almost automorphic mild solutions.

Pseudo almost periodic solutions in the alpha-norm and Stepanov's sense for some evolution equations

UDC 517.9 Our aim is to present the concept of double-measure ergodic and double-measure pseudo almost periodic functions in Stepanov's sense. In addition, we present numerous interesting results, such as the composition theorems and completeness properties for these two spaces of the considered functions. We also establish the existence and uniqueness for the double-measure pseudo almost periodic mild solutions in Stepanov's sense for some evolution equations.

Stability analysis of spinning liquid-filled cylinders with exponentially varying cross section

In this paper, the stability analysis of a spinning liquid-filled cylinder with exponentially varying cross sections is carried out. Based on the spinning Euler–Bernoulli beam theory, the governing equation for vibration of the spinning cylinder is formulated by using Hamilton's principle. The relative perturbation motion equation of the rotating fluid is derived using the composition theorem of acceleration. Combined with the flow boundary conditions, the fluid forces exerted on the cylinder are obtained. Using the analytical method, the characteristic frequency equation of the system is determined. The stability of the considered system is determined by eigenvalue analysis. The accuracy of the proposed model is validated by comparing it with the existing data in the literature. Finally, a detailed parameter study is conducted to demonstrate the effects of mass ratio, cavity ratio, taper parameter, thickness ratio, and axial position on the vibration and stability of the system. The results show that these parameters play an important role in the instability, natural frequency, and critical spinning speed of the spinning taper cylinder partially filled with liquid.

Swellfish privacy: Supporting time-dependent relevance for continuous differential privacy

Today, continuous publishing of differentially private query results is the de-facto standard. However, even today’s most advanced privacy frameworks for streams are not customizable enough to consider that privacy goals of humans change as quickly as human life. We name this time-dependent relevance of privacy goals. Instead, upon design time, one needs to estimate the worst case. Then, one hopes that this protection is sufficient and accepts that one protects against this case all the time, even if it is currently not relevant. Designing a privacy framework being aware of time-dependent relevance implies two effects, which – properly exploited – allow to tune data utility beyond incremental design of a novel privacy mechanism for an existing framework. In this paper, we propose such a new framework, named Swellfish Privacy. We also introduce two tools for designing Swellfish-private mechanisms, namely, time-variant sensitivity and a composition theorem, each implying one effect a mechanism can exploit for improving data utility. In a realistic case study, we show that exploiting both effects improves data utility by one to three orders of magnitude compared to state-of-the-art w-event DP mechanisms. Finally, we generalize the case study by showing how to estimate the strength of the effects for arbitrary use cases.

Stochastic Nicholson’s blowflies model with delays

In this research work, we introduce a new concept of double measure ergodic processes to identify the spaceof double measure pseudo almost periodic (or pap) processes in the pth mean sense. We display some findings regarding the completeness as well as the composition theorems and the invariance of the space consisting in double measure pap processes. Motivated by the above-mentioned results, the Banach fixed point theorem and the stochastic analysis techniques, we prove the existence, uniqueness and the global exponential stability of doubly measure pseudo almost periodic mild solution for a class of stochastic Nicholson’s blowflies model with mixed delays in a separable real Hilbert space. The concluding part of the work is crowned with an example to confirm the reliability and feasibility of our findings.

Besicovitch Almost Periodic Solutions of Abstract Semi-Linear Differential Equations with Delay

In this paper, first, we give a definition of Besicovitch almost periodic functions by using the Bohr property and the Bochner property, respectively; study some basic properties of Besicovitch almost periodic functions, including composition theorem; and prove the equivalence of the Bohr definition and the Bochner definition. Then, using the contraction fixed point theorem, we study the existence and uniqueness of Besicovitch almost periodic solutions for a class of abstract semi-linear delay differential equations. Even if the equation we consider degenerates into ordinary differential equations, our result is new.

Almost Automorphic Solutions in the Sense of Besicovitch to Nonautonomous Semilinear Evolution Equations

In this paper, we study the existence of almost automorphic solutions in the sense of Besicovitch for a class of semilinear evolution equations. Firstly, we study some basic properties of almost automorphic functions in the sense of Besicovitch, including the composition theorem. Then, by using the Banach fixed point theorem, the existence of almost automorphic solutions in the sense of Besicovitch to the semilinear equation is obtained. Finally, we give an example of partial differential equations to illustrate the applicability of our results.

Pseudo compact almost automorphy of neutral type Clifford-valued neural networks with mixed delays

<p style='text-indent:20px;'>We consider a class of neutral type Clifford-valued cellular neural networks with discrete delays and infinitely distributed delays. Unlike most previous studies on Clifford-valued neural networks, we assume that the self feedback connection weights of the networks are Clifford numbers rather than real numbers. In order to study the existence of <inline-formula><tex-math id="M1">\begin{document}$ (\mu, \nu) $\end{document}</tex-math></inline-formula>-pseudo compact almost automorphic solutions of the networks, we prove a composition theorem of <inline-formula><tex-math id="M2">\begin{document}$ (\mu, \nu) $\end{document}</tex-math></inline-formula>-pseudo compact almost automorphic functions with varying deviating arguments. Based on this composition theorem and the fixed point theorem, we establish the existence and the uniqueness of <inline-formula><tex-math id="M3">\begin{document}$ (\mu, \nu) $\end{document}</tex-math></inline-formula>-pseudo compact almost automorphic solutions of the networks. Then, we investigate the global exponential stability of the solution by employing differential inequality techniques. Finally, we give an example to illustrate our theoretical finding. Our results obtained in this paper are completely new, even when the considered networks are degenerated into real-valued, complex-valued or quaternion-valued networks.</p>

Differential Privacy for Deep and Federated Learning: A Survey

Differential privacy (DP) is considered a de-facto standard for protecting users' privacy in data analysis, machine, and deep learning. Existing DP-based privacy-preserving training approaches consist of adding noise to the clients' gradients before sharing them with the server. However, implementing DP on the gradient is not efficient as the privacy leakage increases by increasing the synchronization training epochs due to the composition theorem. Recently researchers were able to recover images used in the training dataset using Generative Regression Neural Network (GRNN) even when the gradient was protected by DP. In this paper, we propose two layers of privacy protection approach to overcome the limitations of the existing DP-based approaches. The first layer reduces the dimension of the training dataset based on Hensel's Lemma. We are the first to use Hensel's Lemma for reducing the dimension (i.e., compress) of a dataset. The new dimensionality reduction method allows reducing the dimension of a dataset without losing information since Hensel's Lemma guarantees uniqueness. The second layer applies DP to the compressed dataset generated by the first layer. The proposed approach overcomes the problem of privacy leakage due to composition by applying DP only once before the training; clients train their local model on the privacy-preserving dataset generated by the second layer. Experimental results show that the proposed approach ensures strong privacy protection while achieving good accuracy. The new dimensionality reduction method achieves an accuracy of 97%, with only 25 % of the original data size.

Besicovitch almost periodic solutions for fractional‐order quaternion‐valued neural networks with discrete and distributed delays

In this paper, we study a class of fractional‐order quaternion‐valued neural networks with discrete and distributed delays by direct method. We first investigate some properties of Besicovitch almost periodic functions, including the composition theorem with deviating arguments. Then, we obtain the existence and uniqueness of Besicovitch almost periodic solutions for such class of networks by considering an appropriate Banach space and using the contraction mapping principle. Furthermore, by making use of a generalized Gronwall inequality, we gain the finite‐time stability of the Besicovitch almost periodic solution. Even if the network under consideration degenerates into a real‐valued or complex‐valued networks, the results of this paper are still brand new. Finally, we use a numerical example to show the feasibility of our theoretical results.

Making the Most of Parallel Composition in Differential Privacy

Abstract We show that the ‘optimal’ use of the parallel composition theorem corresponds to finding the size of the largest subset of queries that ‘overlap’ on the data domain, a quantity we call the maximum overlap of the queries. It has previously been shown that a certain instance of this problem, formulated in terms of determining the sensitivity of the queries, is NP-hard, but also that it is possible to use graph-theoretic algorithms, such as finding the maximum clique, to approximate query sensitivity. In this paper, we consider a significant generalization of the aforementioned instance which encompasses both a wider range of differentially private mechanisms and a broader class of queries. We show that for a particular class of predicate queries, determining if they are disjoint can be done in time polynomial in the number of attributes. For this class, we show that the maximum overlap problem remains NP-hard as a function of the number of queries. However, we show that efficient approximate solutions exist by relating maximum overlap to the clique and chromatic numbers of a certain graph determined by the queries. The link to chromatic number allows us to use more efficient approximate algorithms, which cannot be done for the clique number as it may underestimate the privacy budget. Our approach is defined in the general setting of f-differential privacy, which subsumes standard pure differential privacy and Gaussian differential privacy. We prove the parallel composition theorem for f-differential privacy. We evaluate our approach on synthetic and real-world data sets of queries. We show that the approach can scale to large domain sizes (up to 1020000), and that its application can reduce the noise added to query answers by up to 60%.

Asymptotic behavior of solutions of some stochastic integro-differential equations with infinite delay

In this paper, a new concept of square-mean compact asymptotically almost automorphy for stochastic processes is introduced and a composition theorem is also proved. And then, this notion is applied to investigate existence and uniqueness theorems for squaremean compact asymptotically almost automorphic mild solutions to an abstract stochastic integro-differential equation with infinite delay. Moreover, an example is given to illustrate our main results.