Dynamic Proof Research Articles

VESPo: Verified Evaluation of Secret Polynomials (with application to dynamic proofs of retrievability)

Proofs of Retrievability are protocols which allow a Client to store data remotely and to efficiently ensure, via audits, that the entirety of that data is still intact. Dynamic Proofs of Retrievability (DPoR) also support efficient retrieval and update of any small portion of the data. We propose a novel protocol for arbitrary outsourced data storage that achieves both low remote storage size and audit complexity. A key ingredient, that can be also of intrinsic interest, reduces to efficiently evaluating a secret polynomial at given public points, when the (encrypted) polynomial is stored on an untrusted Server. The Server performs the evaluations and also returns associated certificates. A Client can check that the evaluations are correct using the certificates and some pre-computed keys, more efficiently than re-evaluating the polynomial. Our protocols support two important features: the polynomial itself can be encrypted on the Server, and it can be dynamically updated by changing individual coefficients cheaply without redoing the entire setup. Our methods rely on linearly homomorphic encryption and pairings, and our implementation shows good performance for polynomial evaluations with millions of coefficients, and efficient DPoR with terabytes of data. For instance, for a 1TB database, compared to the state of art, we can reduce the Client storage by 5000x, communication size by 20x, and client-side audit time by 2x, at the cost of one order of magnitude increase in server-side audit time.

Modular orbits on the representation spaces of compact abelian Lie groups

Let S be a closed surface of genus g greater than zero. In the present paper, we study the topological-dynamical action of the mapping class group on the \mathbb T^n -character variety giving necessary and sufficient conditions for \mathrm{Mod}(S) -orbits to be dense. As an application, such a characterisation provides a dynamical proof of the Kronecker's theorem concerning inhomogeneous Diophantine approximation.

TB-ICT: A Trustworthy Blockchain-Enabled System for Indoor Contact Tracing in Epidemic Control

Recently, as a consequence of the COVID-19 pandemic, dependence on Contact Tracing (CT) models has significantly increased to prevent spread of this highly contagious virus and be prepared for the potential future ones. Since the spreading probability of the novel coronavirus in indoor environments is much higher than that of the outdoors, there is an urgent and unmet quest to develop/design efficient, autonomous, trustworthy, and secure indoor CT solutions. Despite such an urgency, this field is still in its infancy. The paper addresses this gap and proposes the Trustworthy Blockchain-enabled system for Indoor Contact Tracing (TB-ICT) framework. The TB-ICT framework is proposed to protect privacy and integrity of the underlying CT data from unauthorized access. More specifically, it is a fully distributed and innovative blockchain platform exploiting the proposed dynamic Proof of Work (dPoW) credit-based consensus algorithm coupled with Randomized Hash Window (W-Hash) and dynamic Proof of Credit (dPoC) mechanisms to differentiate between honest and dishonest nodes. The TB-ICT not only provides a decentralization in data replication but also quantifies the node’s behavior based on its underlying credit-based mechanism. For achieving high localization performance, we capitalize on availability of Internet of Things (IoT) indoor localization infrastructures, and develop a data driven localization model based on Bluetooth Low Energy (BLE) sensor measurements. The simulation results show that the proposed TB-ICT prevents the COVID-19 from spreading by implementation of a highly accurate contact tracing model while improving the users’ privacy and security.

Radical theory of Scott-open filters

Following the Kronecker–Duval or D5 philosophy of dynamic evaluation in computer algebra, the dynamical proof method was brought into constructive algebra in order to obtain computational interpretations of the individual, concrete instances of the Kuratowski–Zorn Lemma modern algebra abounds with. We now push this approach into complete lattices, and thus capture the computational content also of generic, abstract forms of the axiom of choice such as the Teichmüller–Tukey lemma.To this end we first strengthen the (key lemma for) the Hofmann–Mislove theorem about Scott-open filters (SOF): to be a member of a SOF is equivalent, with the axiom of choice, to membership of all complete extensions. This then allows for a constructive treatment: membership of the given SOF means that from the potential element one can grow a finite labelled binary tree of an inductively defined type such that every branch of the tree eventually dips into the SOF.The ideal objects characteristic for invocations of the Kuratowski–Zorn Lemma are thus approximated and actually replaced by paths in those trees, by which we shed light on the interaction of transfinite methods and their dynamical interpretation. The first test case includes abstract dependence, especially bases of vector spaces.Key to the above lies in taking Erné's saturation map as a radical operator for Scott-open sets. Through a generalised inductive definition this radical can be pinned down in terms of universal properties; and reaches into classical pointfree topology, as exemplified by a succinct proof of Isbell's spatiality theorem.

Lightweight Mutual Authentication Scheme Enabled by Stateless Blockchain for UAV Networks

The UAV network composed of resource-constrained lightweight UAV swarms can efficiently accomplish mission with time critical requirements in dynamic and complex environments. However, the trusted authentication of network nodes poses a huge challenge due to its own resource constraints, the lack of trusted centralized support, frequent joining or departure of UAVs to or from the network, and the presence of cyber-attacks. In this paper, we propose a stateless blockchain based on triple aggregatable subvector commitment and present a dynamic proof of trust authorization consensus mechanism with a periodic random selection of authorized nodes to guarantee the trustworthiness of mutual authentication of UAV nodes. Our proposed triple vector authentication solution solves several of the challenges mentioned above very well. The extensive experiments demonstrate that our blockchain-based authentication scheme enjoins significant advantages over the four schemes currently available for UAV network authentication in terms of single authentication latency, speed of energy consumption, average computational cost, and end-to-end latency.

A dynamical proof of the van der Corput inequality

We provide a dynamical proof of the van der Corput inequality for sequences in Hilbert spaces that is based on the Furstenberg correspondence principle. This is done by reducing the inequality to the mean ergodic theorem for contractions on Hilbert spaces. The key difficulty therein is that the Furstenberg correspondence principle is, a priori, limited to scalar-valued sequences. We, therefore, discuss how interpreting the Furstenberg correspondence principle via the Gelfand–Naimark–Segal construction for -algebras allows to study not just scalar but general Hilbert space-valued sequences in terms of unitary operators. This yields a proof of the van der Corput inequality in the spirit of the Furstenberg correspondence principle and the flexibility of this method is discussed via new proofs for different variants of the inequality.

Dynamic proof of retrievability based on public auditing for coded secure cloud storage

Dynamic proof of retrievability based on public auditing for coded secure cloud storage

A Dynamical Proof of the Prime Number Theorem

We present a new, elementary, dynamical proof of the prime number theorem.

A new dynamical proof of the Shmerkin–Wu theorem

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ a < b $\end{document}</tex-math></inline-formula> be multiplicatively independent integers, both at least <inline-formula><tex-math id="M2">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>. Let <inline-formula><tex-math id="M3">\begin{document}$ A,B $\end{document}</tex-math></inline-formula> be closed subsets of <inline-formula><tex-math id="M4">\begin{document}$ [0,1] $\end{document}</tex-math></inline-formula> that are forward invariant under multiplication by <inline-formula><tex-math id="M5">\begin{document}$ a $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ b $\end{document}</tex-math></inline-formula> respectively, and let <inline-formula><tex-math id="M7">\begin{document}$ C : = A\times B $\end{document}</tex-math></inline-formula>. An old conjecture of Furstenberg asserted that any planar line <inline-formula><tex-math id="M8">\begin{document}$ L $\end{document}</tex-math></inline-formula> not parallel to either axis must intersect <inline-formula><tex-math id="M9">\begin{document}$ C $\end{document}</tex-math></inline-formula> in Hausdorff dimension at most <inline-formula><tex-math id="M10">\begin{document}$ \max\{\dim C,1\} - 1 $\end{document}</tex-math></inline-formula>. Two recent works by Shmerkin and Wu have given two different proofs of this conjecture. This note provides a third proof. Like Wu's, it stays close to the ergodic theoretic machinery that Furstenberg introduced to study such questions, but it uses less substantial background from ergodic theory. The same method is also used to re-prove a recent result of Yu about certain sequences of sums.</p>

Dynamical Approach to the TAP Equations for the Sherrington–Kirkpatrick Model

We present a new dynamical proof of the Thouless-Anderson-Palmer (TAP) equations for the classical Sherrington-Kirkpatrick spin glass at sufficiently high temperature. In our derivation, the TAP equations are a simple consequence of the decay of the two point correlation functions. The methods can also be used to establish the decay of higher order correlation functions. We illustrate this by proving a suitable decay bound on the three point functions from which we derive an analogue of the TAP equations for the two point functions.

CAPD::DynSys: A flexible C++ toolbox for rigorous numerical analysis of dynamical systems

We present the CAPD::DynSys library for rigorous numerical analysis of dynamical systems. The basic interface is described together with several interesting case studies illustrating how it can be used for computer-assisted proofs in dynamics of ODEs.

Adaptive nonsingular proportional–integral–derivative-type terminal sliding mode tracker based on rapid reaching law for nonlinear systems

This article deals with a novel adaptive robust controller for uncertain nonlinear systems relying on a proportional–integral–derivative-type nonsingular fast terminal sliding mode control. In this nonsingular proportional–integral–derivative-type terminal sliding mode controller nonsingular fast terminal sliding mode control, the nonsingular fast terminal sliding mode control sliding surface is modified with integral to match with the proportional–integral–derivative-type structure to obtain the essential attributes, namely, quick transient response, finite-time convergence, negligible steady-state error, and chattering cancellation. Furthermore, a novel rapid reaching law is also suggested with dynamic proof for providing the robustness during transient phase. The controller stability and convergence is mathematically analyzed using the Lyapunov theory. The overall control structure is simulated on MATLAB® software and tested for trajectory tracking of a two-degree-of-freedom revolute–prismatic joint industrial robotic manipulator. The rigorous test results show the performance efficacy of the innovative controller.

Dynamic Proof of Data Possession and Replication With Tree Sharing and Batch Verification in the Cloud

Cloud storage attracts a lot of clients to join the paradise. For a high data availability, some clients require their files to be replicated and stored on multiple servers. Because clients are generally charged based on the redundancy level required by them, it is critical for clients to obtain convincing evidence that all replicas are stored correctly and are updated to the up-to-date version. In this article, we propose a dynamic proof of data possession and replication (DPDPR) scheme, which is proved to be secure in the defined security model. Our scheme shares a single authenticated tree across multiple replicas, which reduces the tree's storage cost significantly. Our scheme allows for batch verification for multiple challenged leaves and can verify multiple replicas in a single batch way, which considerably save bandwidth and computation resources during audit process. We also evaluate the DPDPR's performance and compare it with the most related scheme. The evaluation results show that our scheme saves almost 66 percent tree's storage cost for three replicas, and obtains almost 60 and 80 percent efficiency improvements in terms of the overall bandwidth and computation costs, respectively, when three replicas are checked and each challenged with 460 blocks.

Dynamics of isolated left orders

A left order of a countable group $G$ is called isolated if it is an isolated point in the compact space $LO(G)$ of all the left orders of $G$. We study properties of a dynamical realization of an isolated left order. Especially we show that it acts on $\mathbb{R}$ cocompactly. As an application, we give a dynamical proof of the Tararin theorem which characterizes those countable groups which admit only finitely many left orders. We also show that the braid group $B_3$ admits countably many isolated left orders which are not the automorphic images of the others.

Irreversibility, the time arrow and a dynamical proof of the second law of thermodynamics

We provide a dynamical proof of the second law of thermodynamics, along the lines of an argument of Penrose and Gibbs, making crucial use of the upper semicontinuity of the mean entropy proved by Robinson and Ruelle and Lanford and Robinson. An example is provided by a class of models of quantum spin systems introduced by Emch and Radin. Consequences regarding irreversibility and the time arrow, as well as possible extensions to quantum continuous systems, are discussed.

Global exponential stabilization for chaotic brushless DC motor with simpler controllers

In this paper, computer assisted proofs in dynamics (CAPD) group and conventional dynamic analysis methods are applied to find the parameters that make the brushless DC motor (BLDCM) system chaotic. For the purpose to make the BLDCM system exponentially stable, four simple controllers with just one state variable are proposed, which means that it only needs one sensor instead of two or more in practice. Numerical simulations show that the motor with the new controllers can converge efficiently. Finally, the topological horseshoe theory is introduced to verify the existence of chaos in BLDCM system.

Logical argumentation by dynamic proof systems

In this paper we provide a proof theoretical investigation of logical argumentation, where arguments are represented by sequents, conflicts between arguments are represented by sequent elimination rules, and deductions are made by dynamic proof systems extending standard sequent calculi. The idea is to imitate argumentative movements in which certain claims are introduced or withdrawn in the presence of counter-claims. This is done by a dynamic evaluation of sequences of sequents, in which the latter are considered ‘derived’ or ‘not derived’ according to the content of the sequence. We show that decisive conclusions of such a process correspond to well-accepted consequences of the underlying argumentation framework. The outcome is therefore a general and modular proof-theoretical approach for paraconsistent and non-monotonic reasoning with argumentation systems.

Random characteristics for Wigner matrices

We extend the random characteristics approach to Wigner matrices whose entries are not required to have a normal distribution. As an application, we give a simple and fully dynamical proof of the weak local semicircle law in the bulk.

Dynamic Proofs of Retrievability for Coded Cloud Storage Systems

Cloud storage allows users to store their data in a remote server to get rid of expensive local storage and management costs and then access data of interest anytime anywhere. A number of solutions have been proposed to tackle the verification of remote data integrity and retrievability in cloud storage systems. Most of existing schemes, however, do not support efficient data dynamics and/or suffer from security vulnerabilities when involving dynamic data operations. In this paper, we propose a dynamic proof of retrievability scheme supporting public auditability and communication-efficient recovery from data corruptions. To this end, we split up the data into data blocks and encode each data block individually using outer code and inner code before outsourcing so that i) an update inside any data block only affects a few codeword symbols and ii) communication-efficient data repair for a breakdown server can be achieved and communication overhead for small data corruptions within a server can be eliminated. Based on the encoded data blocks, we utilize rb23Tree to enforce the data sequence for dynamic operations, preventing the cloud service provider from manipulating data block to pass the integrity check in the dynamic scenario. Formal security analysis and extensive experimental evaluations are conducted, showing that the proposed scheme is practical for use in cloud storage systems.

DeyPoS-homomorphism authenticated tree (HAT) for multi-user locality in cloud storage

The technology PoS (Dynamic Proof of storage) is a cryptographic primordial allows a abuser to test the reliability of subcontracted documents and effectively replace documents in the cloud storage system. Despite the fact that investigators have projected several dynamic proofs of storage designs in distinct client settings, hassle in the multi-client settings have now not been examined adequately. In sensible multi-client cloud server storage space wishes a cozy client part cross client system of deduplication, it permits client toward bypass importing manner as well as gain instantly the rights of the files, while different vendors of the same files hold uploaded to the cloud system server. In the direction of familiarity not a bit of prevailing dynamic Proof of Storages can guide this system. This research article we are bring the model of dynamic proof of storage in deduplicatable system and endorse a green creation known as Dedupicatable Dynamic Proof of Storage (DeyPoS), on the way to attain DeyPoS and comfy reduplication concurrently in cross client. Taking into account confront of formation assortment and personal blot creation make use of a new tool called HAT (Homomorphic Authenticated Tree). Also verify precautions of creation and the hypothetical, investigational outcomes shows that the creation is green in use.