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Verification of cryptocurrency consensus protocols: reenterable colored Petri net model design
... A fifth team uses colored Petri nets (tokens with a data value) for the efficient simulation of blockchain networks, especially those with more than one hundred nodes [Zaitsev, et al., 2024]. Invoking a colored Petri net consensus protocol, the project defines a re-enterable colored Petri net model representing a general construct for model-driven development. ...
Blockchains are formal systems for equipping objects with value, transacting their exchange, and creating domain-specific event histories. Categorical cryptoeconomics is the application of category-theoretic methods to blockchain study with formalisms which pertain to blockchains and generalize to the programmable computational infrastructure more broadly. Section 1 provides an overview of twenty categorical cryptoeconomic primitives (in algebraic topology (persistent cohomology, semitopology), logic, sheaves, set theory, group theory, optics, and blockchain Petri nets) and their use in consensus, ledger construction, mining, and smart contract platforms. Section 2 introduces four progressively higher categorical cryptoeconomic formulations: HoTT (homotopy type theory) blockchains, Petri net computad ledgers, coregulator DAOs (decentralized autonomous organizations), and cohomology ZKPs (zero-knowledge proofs). The progression is first, nodes as themselves simplicial sets, fibrations, and 2-Segal spaces, second, nodes switched as gradients, third, time-modulated node and path propagation, and fourth, physics-agnostic node and path multiplexing. A 2-category of smart network technologies is envisioned with object instances of blockchains, AI, deep learning, robotics, autonomous vehicles, and digital biology health twins, and morphisms as structure-preserving functors.
... Zaitsev et al. [9] updated the colored Petri net model of the A k Proof-of-work agreement protocol presented at [8]. ...
Distributed global snapshot (DGS) is one of the fundamental protocols in distributed systems. It is used for different applications like collecting information from a distributed system and taking checkpoints for process rollback. The Chandy–Lamport protocol (CLP) is famous and well-known for taking DGS. The main aim of this protocol was to generate consistent cuts without interrupting the regular operation of the distributed system. CLP was the origin of many future protocols and inspired them. The first aim of this paper is to propose a novel formal hierarchical parametric colored Petri net model of CLP. The number of constituting processes of the model is parametric. The second aim is to automatically generate a novel message sequence chart (MSC) to show detailed steps for each simulation run of the snapshot protocol. The third aim is model checking of the proposed formal model to verify the correctness of CLP and our proposed colored Petri net model. Having vital tools helps greatly to test the correct operation of the newly proposed distributed snapshot protocol. The proposed model of CLP can easily be used for visually testing the correct operation of the new future under-development DGS protocol. It also permits formal verification of the correct operation of the new proposed protocol. This model can be used as a simple, powerful, and visual tool for the step-by-step run of the CLP, model checking, and teaching it to postgraduate students. The same approach applies to similar complicated distributed protocols.
Partial differential equations and systems with certain boundary conditions specify continuous processes significant for both large-scale simulations in computer-aided design using HPC and subsequent real-time control of embedded applications using dedicated hardware. The paper develops a spectrum of techniques based on a family of place-transition nets aimed at the computing and communication structure design for fast mass-parallel numerical solving of PDEs. For the HPC domain, we develop models of interconnects in the form of infinite nets and graphical programs in the form of Sleptsov nets. For the embedded control domain, we develop specialized lattices for fast numerical solving PDE based on integer number approximation specified with Sleptsov-Salwicki nets to be implemented on dedicated hardware, which we prototype on FPGAs. For mass-parallel solving of PDEs, we employ ad-hoc finite-difference schemes and iteration methods that allow us to recalculate the lattice values in a single time cycle suitable for control of hypersonic objects and thermonuclear reactions.
In traditional mobile networks, trust between subscribers and their serving networks relies on a hardware root-of-trust: the Subscriber Identity Module (SIM). Conversely, trust between service and home networks is established via Trusted Third Parties (TTPs), known as Clearing Houses (CHs). The 6G environment will witness a substantial increase in subscriber numbers, driven by the mass deployment of the Internet of Everything (IoE) and improvements in network performance. Simultaneously, the performance capabilities required of TTPs to manage trustworthy operator-to-operator (O2O) interactions in 6G must align with the demands of the 6G ecosystem. This work focuses on enhancing CH intermediation capabilities to support O2O trustworthy interactions within the 6G context. Given the close connection between performance and trustworthiness, this paper explores these aspects by modeling interactions between communication parties using a Petri Net model. This model is applied to analyze the quantitative relationships among the non-functional requirements of future 6G communication scenarios, considering both traditional and blockchain-based approaches.
We report our experience in the formal verification of the reference implementation of the Beacon Chain. The Beacon Chain is the backbone component of the new Proof-of-Stake Ethereum 2.0 network: it is in charge of tracking information about the validators , their stakes , their attestations (votes) and if some validators are found to be dishonest, to slash them (they lose some of their stakes). The Beacon Chain is mission-critical and any bug in it could compromise the whole network. The Beacon Chain reference implementation developed by the Ethereum Foundation is written in Python, and provides a detailed operational description of the state machine each Beacon Chain’s network participant (node) must implement. We have formally specified and verified the absence of runtime errors in (a large and critical part of) the Beacon Chain reference implementation using the verification-friendly language Dafny. During the course of this work, we have uncovered several issues, proposed verified fixes. We have also synthesised functional correctness specifications that enable us to provide guarantees beyond runtime errors. Our software artefact with the code and proofs in Dafny is available at https://github.com/ConsenSys/eth2.0-dafny .
Almost all real world activities have been digitized and there are various client server architecture based systems in place to handle them. These are all based on trust on third parties. There is an active attempt to successfully implement blockchain based systems which ensures that the IT systems are immutable, double spending is avoided and cryptographic strength is provided to them. A successful implementation of blockchain as backbone of existing information technology systems is bound to eliminate various types of fraud and ensure quicker delivery of the item on trade. To adapt IT systems to blockchain architecture, an efficient consensus algorithm need to be designed. Blockchain based on proof of work first came up as the backbone of cryptocurrency. After this, several other methods with variety of interesting features have come up. In this paper, we conduct a survey on existing attempts to achieve consensus in block chain. A federated consensus method and a proof of validation method are being compared.
The automation task of evaluating the functional characteristics of IP-networks with dynamic routing based on RIP protocol is solved for the
designing networks process. A method of reenterable model construction is proposed, this method provides reuse of models, which are technologyoriented and invariant to the network structure. Measuring fragments in parametric form added to the model of terminal networks for computing experiments and estimating of IP-network QoS parameters. A comparative analysis of IP-network QoS parameters carried out in conditions with low and medium load, and in a normal operating mode and temporarily disable the router ports. An estimation of the useful bandwidth obtained for different types of loads and conditions of the network operation. The experiments revealed that the values of QoS parameters of the tested network do not change significantly during normal operation and temporarily disable of the router ports, change the dispersion value is an
indicator of problems in the network. Automation of performance evaluation promotes for reducing of network design terms.
With their intuitive graphical approach and expressive analysis techniques, Petri nets are suitable for a wide range of applications and teaching scenarios, and they have gained wide acceptance as a modeling technique in areas such as software design and control engineering. The core theoretical principles have been studied for many decades and there is now a comprehensive research literature that complements the extensive implementation experience. In this book the author presents a clear, thorough introduction to the essentials of Petri nets. He explains the core modeling techniques and analysis methods and he illustrates their usefulness with examples and case studies. Part I describes how to use Petri nets for modeling; all concepts are explained with the help of examples, starting with a generic, powerful model which is also intuitive and realistic. Part II covers the essential analysis methods that are specific to Petri nets, introducing techniques used to formulate key properties of system nets and algorithms for proving their validity. Part III presents case studies, each introducing new concepts, properties and analysis techniques required for very different modeling tasks. The author offers different paths among the chapters and sections: the elementary strand for readers who wish to study only elementary nets; the modeling strand for those who wish to study the modeling but not the analysis of systems; and finally the elementary models of the modeling strand for those interested in technically simple, but challenging examples and case studies. The author achieves an excellent balance between consistency, comprehensibility and correctness in a book of distinctive design. Among its characteristics, formal arguments are reduced to a minimum in the main text with many of the theoretical formalisms moved to an appendix, the explanations are supported throughout with fully integrated graphical illustrations, and each chapter ends with exercises and recommendations for further reading. The book is suitable for students of computer science and related subjects such as engineering, and for a broad range of researchers and practitioners. © Springer-Verlag Berlin Heidelberg 2013. All rights are reserved.
Foundations of the functional Petri net theory were developed for the analysis of detailed models of networking protocols. Methods of timed Petri nets reduction on the basis of equivalent transformations of formulas describing their transfer function were constructed; weak types of nets equivalence were investigated. Methods of synthesis of continuous (fuzzy) logic functions given in tabular form were developed. Bases of the theory of clans of systems of the linear algebraic equations were built, allowing the acceleration of the systems solving in rings with a sign, as well as the corresponding algorithms and software. Methods of synthesis of Petri net models on standard specifications of telecommunication protocols with use of intermediate language of cooperating consecutive Hoare processes were developed. Petri net model of the electronic commerce protocol IOTP was synthesized. Verification of protocols ECMA, BGP, TCP, IOTP was fulfilled. The method of compositional calculation of invariants for infinite Petri nets with regular structure was developed.
A new digital signature based only on a conventional encryption function (such as DES) is described which is as secure as
the underlying encryption function -- the security does not depend on the difficulty of factoring and the high computational
costs of modular arithmetic are avoided. The signature system can sign an unlimited number of messages, and the signature
size increases logarithmically as a function of the number of messages signed. Signature size in a ‘typical’ system might
range from a few hundred bytes to a few kilobytes, and generation of a signature might require a few hundred to a few thousand
computations of the underlying conventional encryption function.
The distributed consensus mechanism is the backbone of the rapidly developing blockchain network. Blockchain platforms consume vast amounts of electricity based on the current consensus mechanism of Proof-of-Work (PoW). Here, we point out a different consensus mechanism named Proof-of-Stake (PoS) that can eliminate the extensive energy consumption of the current PoW-based blockchain. We comprehensively elucidate the current and projected energy consumption and carbon footprint of the PoW- and PoS-based Bitcoin and Ethereum blockchain platforms. The model of energy consumption of PoS-based Ethereum blockchain can lead the way toward the prediction of other PoS-based blockchain technologies in the future. With the widespread adoption of blockchain technology, if the current PoW mechanism continues to be employed, the carbon footprint of Bitcoin and Ethereum will push the global temperature above 1.5 °C in this century. However, a PoS-based blockchain can reduce the carbon footprint by 99% compared to the PoW mechanism. The small amount of carbon footprint from PoS-based blockchain could make blockchain an attractive technology in a carbon-constrained future. The study sheds light on the urgency of developing the PoS mechanism to solve the current sustainability problem of blockchain.
Multidimensional torus interconnect finds wide application in modern exascale computing. For models design in high-performance computing, grid and cloud computing, and also systems biology, two basic ways of specifying spatial structures with Petri nets are considered – an infinite Petri net specified by a parametric expression (PE) and a reenterable coloured Petri net (CPN). The paper studies a composition of hypertorus grid models in the form of a PE and a reenterable CPN, their mutual transformations, and unfolding into a place/transition net; the parameters are the number of dimensions and the size of grid. A grid is composed via connection of neighbouring cells by dedicated transitions modelling channels. Reenterable model peculiarities are explained on step-by-step simulation examples. The rules of mutual transformations of Petri net spatial specifications are specified. Comparative investigation of two mentioned forms of spatial specifications is implemented, including analysis techniques and tools. CPNs are convenient for the state space analysis. The main advantage of PEs is the ability to obtain linear invariants and other structural constructs of Petri nets, for instance, siphons and traps, in parametric form that allows us to draw conclusions on Petri net properties for any values of parameters.
Proof-of-work agreement protocol, offered by Keller and Böhme, is analysed by coloured Petri nets and refined. Blockchain technology, based on proof-of-work procedure and Nakomoto consensus negotiations, represents fundamentals of many kinds of cryptocurrency widespread recently. The protocol, called Ak, works in continuous time which is simulated using random exponential distribution function of CPN Tools system, obtained values rounded to map them into discrete time of a coloured Petri net. Hierarchical model consists of an environment subnet and a given number of nodes communicating via an unstructured network represented by a single place; the model of node is further structured based on event handlers of the protocol source specification such as initialisation, activation, message delivering, and termination condition check. Based on the simulation results, modifications of the protocol and its parameters are recommended which improve some imperfections of the protocol.
GPSS (General Purpose Simulation System) is overviewed briefly. The approach taken in GPSS to model a one-line, one-server system is explained, implementation details are provided, and results are discussed. References and suggestions for further study are given.
A purely peer-to-peer version of electronic cash would allow online payments to be sent directly from one party to another without going through a financial institution. Digital signatures provide part of the solution, but the main benefits are lost if a trusted third party is still required to prevent double-spending. We propose a solution to the double-spending problem using a peer-to-peer network. The network timestamps transactions by hashing them into an ongoing chain of hash-based proof-of-work, forming a record that cannot be changed without redoing the proof-of-work. The longest chain not only serves as proof of the sequence of events witnessed, but proof that it came from the largest pool of CPU power. As long as a majority of CPU power is controlled by nodes that are not cooperating to attack the network, they'll generate the longest chain and outpace attackers. The network itself requires minimal structure. Messages are broadcast on a best effort basis, and nodes can leave and rejoin the network at will, accepting the longest proof-of-work chain as proof of what happened while they were gone.
Coloured Petri Nets (CPN) is a graphical language for modelling and validating concurrent and distributed systems, and other systems in which concurrency plays a major role. The development of such systems is particularly challenging because of inherent intricacies like possible nondeterminism and the immense number of possible execution sequences. In this textbook Jensen and Kristensen introduce the constructs of the CPN modelling language and present the related analysis methods in detail. They also provide a comprehensive road map for the practical use of CPN by showcasing selected industrial case studies that illustrate the practical use of CPN modelling and validation for design, specification, simulation, verification and implementation in various application domains. Their presentation primarily aims at readers interested in the practical use of CPN. Thus all concepts and constructs are first informally introduced through examples and then followed by formal definitions (which may be skipped). The book is ideally suitable for a one-semester course at an advanced undergraduate or graduate level, and through its strong application examples can also serve for self-study. An accompanying website offers additional material such as slides, exercises and project proposals.
PATRICIA is an algorithm which provides a flexible means of storing, indexing, and retrieving information in a large file, which is economical of index space and of reindexing time. It does not require rearrangement of text or index as new material is added. It requires a minimum restriction of format of text and of keys; it is extremely flexible in the variety of keys it will respond to. It retrieves information in response to keys furnished by the user with a quantity of computation which has a bound which depends linearly on the length of keys and the number of their proper occurrences and is otherwise independent of the size of the library. It has been implemented in several variations as FORTRAN programs for the CDC-3600, utilizing disk file storage of text. It has been applied to several large information-retrieval problems and will be applied to others.
Two kinds of contemporary developments in cryptography are examined. Widening applications of teleprocessing have given rise to a need for new types of cryptographic systems, which minimize the need for secure key distribution channels and supply the equivalent of a written signature. This paper suggests ways to solve these currently open problems. It also discusses how the theories of communication and computation are beginning to provide the tools to solve cryptographic problems of long standing.
Reliable computer systems must handle malfunctioning components that give conflicting information to different parts of the system. This situation can be expressed abstractly in terms of a group of generals of the Byzantine army camped with their troops around an enemy city. Communicating only by messenger, the generals must agree upon a common battle plan. However, one of more of them may be traitors who will try to confuse the others. The problem is to find an algorithm to ensure that the loyal generals will reach agreement. It is shown that, using only oral messages, this problem is solvable if and only if more than two-thirds of the generals are loyal; so a single traitor can confound two loyal generals. With unforgeable written messages, the problem is solvable for any number of generals and possible traitors. Applications of the solutions to reliable computer systems are then discussed.
Analysis of nakamoto consensus cryptology ePrint archive
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A review of hashing algorithms in cryptocurrency
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Reenterable colored Petri net models of networks grids and clouds: case study for provider backbone bridge
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Parametrical model of MPLS networks in the form of colored Petri nets
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Verification of a merkle patricia tree library using F*
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Combining GHOST and Casper
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Code schemes vary with management goals
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