In leakage resilient cryptography, there is a seemingly inherent restraint on the ability of the adversary that it cannot get access to the leakage oracle after the challenge. Recently, a series of works made a breakthrough to consider a postchallenge leakage. They presented achievable public key encryption (PKE) schemes which are semantically secure against after-the-fact leakage in the split-state model. This model puts a more acceptable constraint on adversary’s ability that the adversary cannot query the leakage of secret states as a whole but the functions of several parts separately instead of prechallenge query only. To obtain security against chosen ciphertext attack (CCA) for PKE schemes against after-the-fact leakage attack (AFL), existing works followed the paradigm of “double encryption” which needs noninteractive zero knowledge (NIZK) proofs in the encryption algorithm. We present an alternative way to achieve AFL-CCA security via lossy trapdoor functions (LTFs) without NIZK proofs. First, we formalize the definition of LTFs secure against AFL (AFLR-LTFs) and all-but-one variants (ABO). Then, we show how to realize this primitive in the split-state model. This primitive can be used to construct AFLR-CCA secure PKE scheme in the same way as the method of “CCA from LTFs” in traditional sense.
Protecting sensitive medical data, including prescription and pill data, during its handling and storage is critical in the digital era. Data vaults that protect privacy provide a strong way to protect this information, guaranteeing that patient information is kept private but yet available for authorised uses. Focussing on the safe preservation of pharmaceutical data, this project investigates the creation of sophisticated algorithms for privacy-preserving data vaults. We start by contrasting the suggested innovative technique, which combines elements of Zero-Knowledge Proofs with Enhanced Homomorphic Encryption, with other known cryptographic and data masking algorithms, such as Differential Privacy, Secure Multi-Party Computation, and Homomorphic Encryption. Data integrity, computational efficiency, and resistance to different attack vectors are some of the characteristics used in the comparison.As the results show, the suggested method offers aimproved performance against confidentiality compromises, especially in real-time data retrieval scenarios, while existing techniques offer varied degrees of efficiency and security. But this comes with more implementation complexity and processing overhead. Improved security characteristics, like less data leakage and strong user authentication systems, are benefits of the suggested approach. Large-scale applications may experience latency problems and require more powerful hardware, which are drawbacks.The trade-offs between various data privacy strategies are highlighted in this study, and it also highlights the necessity for on-going innovation in privacy-preserving technology, which makes a contribution to the region.
With the increasing interest in connected vehicles along with electrification opportunities, there is an ongoing effort to automate the charging process of electric vehicles (EVs) through their capabilities to communicate with the infrastructure and each other. However, charging EVs takes time and thus in-advance scheduling is needed. As this process is done frequently due to limited mileage of EVs, it may expose the locations and charging pattern of the EV to the service providers, raising privacy concerns for their users. Nevertheless, the EV still needs to be authenticated to charging providers, which means some information will need to be provided anyway. While there have been many studies to address the problem of privacy-preserving authentication for vehicular networks, such solutions will be void if charging payments are made through traditional means. In this thesis, we tackle this problem by utilizing distributed applications enabled by Blockchain and smart contracts. We adapt zero-knowledge proofs to Blockchain for enabling privacy-preserving authentication while removing the need for a central authority. We introduce two approaches, one using a token-based mechanism and another utilizing the Pederson Commitment scheme to realize anonymous authentication. We also describe a protocol for the whole process which includes scheduling and charging operations. The evaluation of the proposed approaches indicates that the overhead of this process is affordable to enable real-time charging operations for connected EVs.
This paper presents a system architecture to promote the development of smart transportation systems. Thanks to the use of distributed ledgers and related technologies, it is possible to create, store and share data generated by users through their sensors, while moving. In particular, IOTA and IPFS are used to store and certify data (and their related metadata) coming from sensors or by the users themselves. Ethereum is exploited as the smart contract platform that coordinates the data sharing and provisioning. The necessary privacy guarantees are provided by the usage of Zero Knowledge Proof. We show some results obtained from some use case scenarios that demonstrate how such technologies can be integrated to build novel smart services and to promote social good in user mobility.
Les primitives lossy trapdoor, preuve à divulgation nulle de connaissance et applications Dans cette thèse, nous étudions deux primitives différentes : les lossy trapdoor functions (LTF) et les systèmes de preuve à divulgation nulle de connaissance. Les LTFs sont des familles de fonctions dans lesquelles les fonctions injectives et les fonctions lossy sont calculatoirement indistinguables. Depuis leur introduction, elles se sont avérées utiles pour la construction de diverses primitives cryptographiques. Nous donnons dans cette thèse des constructions efficaces d’une variante de la LTF : le filtre algébrique lossy. Avec cette variante, nous pouvons améliorer l’efficacité du schéma de chiffrement KDM-CCA et extracteur flous. Dans la deuxième partie de cette thèse, nous étudions les constructions de systèmes de preuve à divulgation nullle de connaissance. Nous donnons la première signature d’anneau de taille logarithmique avec la sécurité étroite en utilisant une variante de Groth-Kolhweiz Σ-protocole dans le modèle de l’oracle aléatoire. Nous proposons également une nouvelle construction d’arguments à divulgation nulle de connaissance non-intéractive et à vérifieur désigné (DVNIZK) sous l’hypothèse de réseaux Euclidiens. En utilisant cette nouvelle construction, nous construisons un système de vote basé sur les réseaux Euclidiens dans le modèle standard.
Sina Rafati Niya, Sebastian Allemann, Arik Gabay, Burkhard Stiller
Data leaks and privacy scandals have been a growing concern of the last decade. While most traditional, i.e., centralized, online platforms require users to register with their personal data, they potentially expose the user's identity and data to be used for unintended purposes. This work proposes TradeMap as an integrated architecture, designing and enabling an online end-to-end (e2e) trading market place, while supporting anonymous management features. TradeMap addresses the Swiss Financial Market Supervisory Authority (FINMA) regulations by designing a FINMA-complaint Know Your Customer (KYC) platform. Additionally, TradeMap is based on blockchains and employs Ethereum Smart Contracts (SC). Thus, trust and anonymity between the market place and the KYC system relies on zero knowledge proof-based SCs used for user identification processes. With this management approach proposed, the user authentication is only verified within the KYC platform, providing a legally valid and fully anonymous online trading platform.
Matteo Varvello, Iñigo Querejeta Azurmendi, Antonio Nappa, Panagiotis N. Papadopoulos · 6 authors
Distributed Virtual Private Networks (dVPNs) are new VPN solutions aiming to solve the trust-privacy concern of a VPN's central authority by leveraging a distributed architecture. In this paper, we first review the existing dVPN ecosystem and debate on its privacy requirements. Then, we present VPN0, a dVPN with strong privacy guarantees and minimal performance impact on its users. VPN0 guarantees that a dVPN node only carries traffic it has "whitelisted", without revealing its whitelist or knowing the traffic it tunnels. This is achieved via three main innovations. First, an attestation mechanism which leverages TLS to certify a user visit to a specific domain. Second, a zero knowledge proof to certify that some incoming traffic is authorized, e.g., falls in a node's whitelist, without disclosing the target domain. Third, a dynamic chain of VPN tunnels to both increase privacy and guarantee service continuation while traffic certification is in place. The paper demonstrates VPN0 functioning when integrated with several production systems, namely BitTorrent DHT and ProtonVPN.
The Holy Grail of a decentralised stablecoin is achieved on rigorous\nmathematical frameworks, obtaining multiple advantageous proofs: stability,\nconvergence, truthfulness, faithfulness, and malicious-security. These\nproperties could only be attained by the novel and interdisciplinary\ncombination of previously unrelated fields: model predictive control, deep\nlearning, alternating direction method of multipliers (consensus-ADMM),\nmechanism design, secure multi-party computation, and zero-knowledge proofs.\nFor the first time, this paper proves:\n - the feasibility of decentralising the central bank while securely\npreserving its independence in a decentralised computation setting\n - the benefits for price stability of combining mechanism design, provable\nsecurity, and control theory, unlike the heuristics of previous stablecoins\n - the implementation of complex monetary policies on a stablecoin, equivalent\nto the ones used by central banks and beyond the current fixed rules of\ncryptocurrencies that hinder their price stability\n - methods to circumvent the impossibilities of Guaranteed Output Delivery\n(G.O.D.) and fairness: standing on truthfulness and faithfulness, we reach\nG.O.D. and fairness under the assumption of rational parties\n As a corollary, a decentralised artificial intelligence is able to conduct\nthe monetary policy of a stablecoin, minimising human intervention.\n
Kakuro is a popular logic puzzle, in which a player fills in all empty squares with digits from 1 to 9 so that the sum of digits in each (horizontal or vertical) line is equal to a given number, called a clue, and digits in each line are all different. In 2016, Bultel, Dreier, Dumas, and Lafourcade proposed a physical zero-knowledge proof protocol for Kakuro using a deck of cards; their proposed protocol enables a prover to convince a verifier that the prover knows the solution of a Kakuro puzzle without revealing any information about the solution. One possible drawback of their protocol would be that the protocol is not perfectly extractable, implying that a prover who does not know the solution can convince a verifier with a small probability; therefore, one has to repeat the protocol to make such an error become negligible. In this paper, to overcome this, we design zero-knowledge proof protocols for Kakuro having perfect extractability property. Our improvement relies on the ideas behind the copy protocols in the field of card-based cryptography. By executing our protocols with a real deck of physical playing cards, humans can practically perform an efficient zero-knowledge proof of knowledge for Kakuro.
Democratic principles demand that every voter should be able to individually verify that their vote is recorded as intended and counted as recorded, without having to trust any authorities. However, most end-to-end (E2E) verifiable voting protocols that provide universal verifiability and voter secrecy implicitly require to trust some authorities or auditors for the correctness guarantees that they provide.
In this paper, we explore the notion of individual verifiability. We evaluate the existing E2E voting protocols and propose a new protocol that guarantees such verifiability without any trust requirements. Our construction depends on a novel vote commitment scheme to capture voter intent that allows voters to obtain a direct zero-knowledge proof of their vote being recorded as intended. We also ensure protection against spurious vote injection or deletion post eligibility verification, and polling-booth level community profiling.
This paper introduces a new capability for group signatures called message-dependent opening . It is intended to weaken the high trust placed on the opener; i.e., no anonymity against the opener is provided by an ordinary group signature scheme. In a group signature scheme with message-dependent opening (GS-MDO), in addition to the opener, we set up an admitter that is not able to extract any user’s identity but admits the opener to open signatures by specifying messages where signatures on the specified messages will be opened by the opener. The opener cannot extract the signer’s identity from any signature whose corresponding message is not specified by the admitter. This paper presents formal definitions of GS-MDO and proposes a generic construction of it from identity-based encryption and adaptive non-interactive zero-knowledge proofs. Moreover, we propose two specific constructions, one in the standard model and one in the random oracle model. Our scheme in the standard model is an instantiation of our generic construction but the message-dependent opening property is bounded. In contrast, our scheme in the random oracle model is not a direct instantiation of our generic construction but is optimized to increase efficiency and achieves the unbounded message-dependent opening property. Furthermore, we also demonstrate that GS-MDO implies identity-based encryption, thus implying that identity-based encryption is essential for designing GS-MDO schemes.
Samad M. E. Sepasgozar, Reyhaneh Karimi, Sara Shirowzhan, Mohammad Mojtahedi · 6 authors
Delay is one of the main challenges of construction projects, and there is still much to overcome in order to reach near zero delay in all construction projects. This project aims to conduct a systematic critical review including a bibliography analysis on delay literature in construction. The main questions consider what has been learnt from a decade investigating delay causes and effects in the construction literature and what factors have been missed in the literature. This paper also presents a new and challenging question regarding how digital tools and associated technologies may prevent any delay in construction projects, which can change the research direction from delay investigations to identifying prevention factors. The paper identifies the delay dataset, including 493 papers investigating delay in construction, and establishes a specific dataset of papers focusing on delay effects and causes (DEC), including 94 selected papers covering different factors examined in over 29 countries such as Iran, India, Turkey, Bangladesh, Saudi Arabia, the United Arab Emirates (UAE), Cambodia, Oman, Malaysia, Taiwan, China, Vietnam, the US, the UK, and Egypt. In addition, the paper identifies 30 critical factors with the frequency of occurrences over three times in the DEC dataset and computes their medians of ranking. This paper also discusses digital tools and methods that can be used for delay analysis and preventions, including MS Project, Oracle Primavera P6, and Open Plan by Deltek. The paper discusses the project schedule delay analysis from project management methodology perspectives. It also discusses the current method’s limitations and future directions, which are based on the identification of the deficiency areas. In total, four overlooked factors are identified and suggested, including faulty data analysis, unmatched structure of the research questionnaires with new knowledge and standards [e.g., Project Management Body of Knowledge (PMBOK)], overlooked effects of digital technologies [e.g., Digital twin, Navisworks, Building Information Model (BIM), Geographic Information System (GIS), and Integrated Project Delivery (IPD)], and ignored job-site technologies. In addition, the paper presents the DEC model for future studies, including four main key factors. These factors are resources (e.g., project budgets, labour, material, equipment, and digital tool), project context, stakeholders performance (e.g., owner/client, consultant/designer, contractor, vendor/supplier), and external factors (e.g., ground condition, site location, regulation, natural disaster), which may significantly affect delay prevention and should be concurrently considered in the future delay investigations, since they may be required for designing an effective mitigation strategy when these proof points are identified. This would significantly help to utilise digital systems to prevent time overruns in different construction contexts.
A primary concern of a wireless sensor network (WSN) is to gather data from the immediate environment of it sensors while minimizing the use of limited network and computational resources. Several studies have focused on how to efficiently store and process sensed data in WSN. Generally, the appropriate method to store sensed data depends on the application for which the WSN was deployed. No matter the application, data collection appears to be a primary function of a WSN. The execution of this function must be coordinated and effective in order to provide WSN with current security standards such as privacy, data integrity and end entity authentication between communicating peers. In this paper, we propose an efficient authentication-based security scheme for data retrieval in WSN. This security scheme combines zero-knowledge proofs (ZKP) and pre-shared key method to provide secured and authenticated communication during data retrieval by a mobile sink in WSN. The security mechanism proposed works on a clustered network topology with an index-based data dissemination scheme. The network employs the concept of Connected Dominating Set (CDS) to form storage and index node sets. Upon a successful peer authentication, the index, located on the index node, is used for efficient retrieval of data. The scheme also provides end-to-end confidentiality given that data is being encrypted before transferred and can be decrypted only at the base station. Security and performance analysis of the proposed scheme show that it addresses all of the aforementioned issues while also satisfying zero-knowledge proofs properties. It is also suitable for devices with limited computational resources as the network can fulfil the purpose of data collection and can be deployed in large-scale wireless sensor networks.
This study investigates the role and functionality of special nucleotide sequences (DNA signatures) to detect the presence of an organism and to distinguish it from all others. After highlighting vulnerabilities of the prevalent DNA signature paradigm for the identification of agricultural genetically modified (GM) organisms it will be argued that these so-called signatures really are no signatures at all - when compared to the notion of traditional (handwritten) signatures and their generalizations in the modern (digital) world. It is suggested that a recent contamination event of an unauthorized GM Bacillus subtilis strain (Paracchini et al. (2017)) in Europe could have been - or the same way could be - the consequence of exploiting gaps of prevailing DNA signatures. Moreover, a recent study (Mueller (2019)) proposes that such DNA signatures may intentionally be exploited to support the counterfeiting or even weaponization of GM organisms (GMOs). These concerns mandate a re-conceptualization of how DNA signatures need to be realized. After identifying central issues of the new vulnerabilities and overlying them with practical challenges that bio-cyber hackers would be facing, recommendations are made how DNA signatures may be enhanced. To overcome the core problem of signature transferability in bioengineered mediums, it is necessary that the identifier needs to remain secret during the entire verification process. On the other hand, however, the goal of DNA signatures is to enable public verifiability, leading to a paradoxical dilemma. It is shown that this can be addressed with ideas that underlie special cryptographic signatures, in particular those of ‘zero-knowledge’ and ‘invisibility.’ This means more than mere signature hiding, but relies on a knowledge-based proof and differentiation of a secret (here, as assigned to specific clones) which can be realized without explicit demonstration of that secret. A reconceptualization of these principles can be used in form of a combined (digital and physical) method to establish confidentiality and prevent un-impersonation of the manufacturer. As a result, this helps mitigate the circulation of possibly hazardous GMO counterfeits and also addresses the situation whereby attackers try to blame producers for deliberately implanting illicit adulterations hidden within authorized GMOs.
Recent advances in the cryptographic field of "Zero-Knowledge Proofs" have sparked a new wave of research, giving birth to many exciting theoretical approaches in the last few years. Such research has often overlapped with the need for private and scalable solutions of Blockchain-based communities, resulting in the first practical implementations of such systems. Many of these innovative constructions have developed in parallel, using different terminologies and evolving into a fragmented ecosystem, calling for their consolidation into the more stable domain of "Verifiable Computation". In this master thesis I propose a unifying Verifiable Computation model for the simplification and efficient comparison of all cryptographic proof systems. I take advantage of this model to analyse innovative technologies (Homomorphic Authenticators, Verifiable Delay Functions) which developed into their own specialised domains, and I attempt to make them more accessible for newcomers to the field. Furthermore, I expand on the future of Verifiable Computation, Universal proof compilers and "Proofs of All", by approaching the state-of-the-art zk-STARK construction from a more accessible and informal design perspective.
E-cash has its merits comparing with other payment modes. However, there are two problems, which are how to achieve practical/complete tracing and how to achieve it in compact E-cash. First, the bank and the TTP (i.e., trusted third party) have different duties and powers in the reality. Therefore, double-spending tracing is bank's task, while unconditional tracing is TTP's task. In addition, it is desirable to provide lost-coin tracing before they are spent by anyone else. Second, compact E-cash is an efficient scheme, but tracing the coins from double-spender without TTP results in poor efficiency. To solve the problems, we present a compact E-cash scheme. For this purpose, we design an embedded structure of knowledge proof based on a new pseudorandom function and improve the computation complexity from O(k) to O(1). Double-spending tracing needs leaking dishonest users' secret knowledge, but preserving the anonymity of honest users needs zero-knowledge property, and our special knowledge proof achieves it with complete proofs. Moreover, the design is also useful for other applications, where both keeping zero-knowledge and leaking information are necessary.
From hairbrushes to scales, all devices have sensors embedded in them to collect and communicate data. Smart Healthcare is proving to be an exciting and dynamic area with lots of room for new innovations and the increasing consumer demand for proactive health monitoring devices. Having India poised to spend a lot on healthcare, recent innovations using IoT devices and big data analytics can propel the healthcare industry into the future. Smart healthcare providers are leveraging cloud computing with fog computing to optimize their healthcare services. These smart healthcare applications depend mainly on the raw sensor data collected, aggregated, and analyzed by the smart sensors. Smart sensors these days generate myriad amount of data like text, image, audio, and video that require real-time or batch processing. Aggregating these diverse data from various types of resources remains a dispute till date. To resolve this issue, we have proposed a softwarized infrastructure that integrates cloud computing and fog computing, message brokers, and Tor for supple, safe, viable, and a concealed IoT exploitation for smart healthcare applications and services. Our proposed platform employs machine-to-machine (M2M) messaging, data fusion and decision fusion, and uses rule-based beacons for seamless data management. Our proposed flexBeacon system provides an IoT infrastructure that is nimble, secure, flexible, private, and reasonable. We have also proposed an M2M transceiver and microcontroller for flawless data incorporation of smart healthcare applications and services. Based on the IoT devices’ technical capabilities and resource availability, some systems are capable of making use of homomorphic encryption and zero knowledge proofs. The proposed flexBeacon platform offers seamless management and data aggregation without loss of accuracy. The cost of implementing a softwarized IoT for smart healthcare is also greatly reduced.
User authentication can rely on various factors (e.g., a password, a cryptographic key, biometric data) but should not reveal any secret or private information. This seemingly paradoxical feat can be achieved through zero-knowledge proofs. Unfortunately, naive password-based approaches still prevail on the web. Multi-factor authentication schemes address some of the weaknesses of the traditional login process, but generally have deployability issues or degrade usability even further as they assume users do not possess adequate hardware. This assumption no longer holds: smartphones with biometric sensors, cameras, short-range communication capabilities, and unlimited data plans have become ubiquitous. In this paper, we show that, assuming the user has such a device, both security and usability can be drastically improved using an augmented password-authenticated key agreement (PAKE) protocol and message authentication codes.
Tamadun Malayonesia yang sarjananya menggunakan bahasa Melayu sebagai pengungkap kecendekiawanannya, masih belum diiktiraf kewujudannya oleh sarjana Barat kerana ketiadaan bukti bertulis pencapaiannya yang tinggi dalam KeJuTSAMA/STEM. Dalam usaha memperbaiki keadaan inilah maka kami telah melakukan penyelidikan, ekskavasi ilmu, sejak 1990-an bagi memperoleh unsur-unsur sains dan matematik (SAMA) dalam bahasa Melayu yang asli dan terkehadapan daripada bangsa lain sezamannya terutamanya Inggeris. Di sini dipaparkan beberapa hasil penyelidikan yang mutakhir tentang perkara tersebut, khususnya untuk bidang angka. Ditunjukkan bahawa bangsa Malayonesia berbahasa Melayu adalah pencipta terawal di dunia sistem angka perpuluhan seperti sekarang ini, sekaligus menjadikan bangsa pencipta angka kosong yang tertua di dunia. Hal ini bertentangan dengan kepercayaan dunia selama ini bahawa sistem angka perpuluhan dengan kosong adalah ciptaan tamadun Hindu, Islam atau Khmer Lama. The existence of the Malayonesian civilization in which her scholars used Malay as the their intellectual expresions, has not yet been recognised by Western sholars because of lack of written proof in her achievement in STEM. In an effort to improve the situation, we have done some knowledge excavation since 1990s in order to obtain original elements of science in Malay language which were ahead of their counterpart in different countries in Europe especially UK at the same period of time. The latest results are presented here, particularly in the fields of numerals. It is shown that the Malayonesian numeral decimal system with her own zero symbol is the oldest in the world. This contradicts the present belief that the decimal numeral system was invented by the Hindu, the Islamic or the Old Khmer civilisation.
Eduardo Morais, Tommy Koens, Cees van Wijk, Aleksei Koren
In last years, there has been an increasing effort to leverage Distributed Ledger Technology (DLT), including blockchain. One of the main topics of interest, given its importance, is the research and development of privacy mechanisms, as for example is the case of Zero Knowledge Proofs (ZKP). ZKP is a cryptographic technique that can be used to hide information that is put into the ledger, while still allowing to perform validation of this data. In this work we describe different strategies to construct Zero Knowledge Range Proofs (ZKRP), as for example the scheme proposed by Boudot in 2001; the one proposed in 2008 by Camenisch et al, and Bulletproofs, proposed in 2017. We also compare these strategies and discuss possible use cases. Since Bulletproofs is the most efficient construction, we will give a detailed description of its algorithms and optimizations. Bulletproofs is not only more efficient than previous schemes, but also avoids the trusted setup, which is a requirement that is not desirable in the context of Distributed Ledger Technology (DLT) and blockchain. In case of cryptocurrencies, if the setup phase is compromised, it would be possible to generate money out of thin air. Interestingly, Bulletproofs can also be used to construct generic Zero Knowledge Proofs (ZKP), in the sense that it can be used to prove generic statements, and thus it is not only restricted to ZKRP, but it can be used for any kind of Proof of Knowledge (PoK). Hence Bulletproofs leads to a more powerful tool to provide privacy for DLT. Here we describe in detail the algorithms involved in Bulletproofs protocol for ZKRP. Also, we present our implementation, which was open sourced.
We consider two-dimensional zero-temperature systems of $N$ particles to which we associate an energy of the form $$ \mathcal{E}[V](X):=\sum_{1\le i<j\le N}V(|X(i)-X(j)|), $$ where $X(j)\in\mathbb R^2$ represents the position of the particle $j$ and $V(r)\in\mathbb R$ is the {pairwise interaction} energy potential of two particles placed at distance $r$. We show that under suitable assumptions on the single-well potential $V$, the ground state energy per particle converges to an explicit constant $\bar{\mathcal E}_{\mathrm{sq}}[V]$ which is the same as the energy per particle in the square lattice infinite configuration. We thus have $$ N{\bar{\mathcal E}_{\mathrm{sq}}[V]}\le \min_{X:\{1,\ldots,N\}\to\mathbb R^2}\mathcal E[V](X)\le N{\bar{\mathcal E}_{\mathrm{sq}}[V]}+O(N^{\frac 1 2}). $$ Moreover $\bar{\mathcal E}_{\mathrm{sq}}[V]$ is also re-expressed as the minimizer of a four point energy. In particular, this happen{s} if the potential $V$ is such that $V(r)=+\infty$ for $r<1$, $V(r)=-1$ for $r\in [1,\sqrt{2}]$, $V(r)=0$ if $r>\sqrt{2}$, in which case ${\bar{\mathcal E}_{\mathrm{sq}}[V]}=-4$. To the best of our knowledge, this is the first proof of crystallization to the square lattice for a two-body interaction energy.
In this paper, we propose a system for mixing transactions in payment networks such as credit networks. Credit networks like Ripple and Stellar are increasingly popular, and can facilitate crosscurrency transactions in a fraction of the time it would take for banks or other financial institutions to process the same transaction, and at a fraction of the cost. Unlike for cryptocurrencies, there has been little work in the area of designing secure and private mixers for credit networks. Mixers for cryptocurrencies such as Bitcoin cannot be directly applied to the credit network domain because credit networks have an inherently different structure and purpose than cryptocurrencies. We design a system that uses cryptographic constructs such as ring signatures, commitments, and zero knowledge proofs to provide security/integrity of all transactions, ensures privacy of the users involved in a transaction, as well as privacy of the amount transacted. We also provide preliminary experimental results.