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- A Zero-Knowledge Proof: Improving Privacy on a Blockchain
- An authentication protocol based on chaos and zero knowledge proof
- Verifiable Credentials Data Model 1.0
- A Survey of Noninteractive Zero Knowledge Proof System and Its Applications

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## A Zero-Knowledge Proof: Improving Privacy on a Blockchain

Zero knowledge proof system which has received extensive attention since it was proposed is an important branch of cryptography and computational complexity theory. Thereinto, noninteractive zero knowledge proof system contains only one message sent by the prover to the verifier. It is widely used in the construction of various types of cryptographic protocols and cryptographic algorithms because of its good privacy, authentication, and lower interactive complexity.

This paper reviews and analyzes the basic principles of noninteractive zero knowledge proof system, and summarizes the research progress achieved by noninteractive zero knowledge proof system on the following aspects: the definition and related models of noninteractive zero knowledge proof system, noninteractive zero knowledge proof system of NP problems, noninteractive statistical and perfect zero knowledge, the connection between noninteractive zero knowledge proof system, interactive zero knowledge proof system, and zap, and the specific applications of noninteractive zero knowledge proof system.

This paper also points out the future research directions. In , Goldwasser et al. The most attractive feature of zero knowledge proof lies in its seemingly contradictory unique nature that a prover can prove the correctness of an assertion to the verifier without leaking any extra information.

It can force the malicious participants in cryptographic protocol to execute in accordance with predetermined steps to ensure the safety of the protocol. Thus it has a broad application prospect. To speak vividly, a verifier who receives the zero knowledge proof of a statement is supposed to be told by God that it is true.

The main features of zero knowledge proof system include completeness, soundness, and zero knowledge. Zero Knowledge. No malicious verifier can get any extra information from the proof procedure, except the correctness of the statement.

Blum et al. Noninteractive zero knowledge proof system contains only a message sent by a prover to verifier, which can be better used in the construction of cryptographic protocols. Thereafter, researches on the theory and applications of NIZK proof system have started successively, including NIZK proof of NP problems and noninteractive statistical perfect zero knowledge as well as the application of NIZK proof to CCA security encryption scheme, anonymous authentication, and the construction of group and ring signature.

In recent years, Groth et al. While for the verifier, zero knowledge means no malicious verifier is able to derive extra knowledge from the process of interaction. In addition, according to different computational capabilities of the prover and verifier, the above properties 2 and 3 can also be modified, respectively. If the indistinguishability of the two probability ensembles in property 3 is statistically indistinguishable or identically distributed, zero knowledge will be correspondingly defined as statistical zero knowledge and perfect zero knowledge.

On the other hand, if soundness holds for any probabilistic polynomial time prover, that is, computational soundness, then the interactive proof system is called the zero knowledge argument system [ 18 ]. At present, it is generally accepted by the researchers to construct NIZK proof system in the common reference string hereinafter referred to as CRS model.

For a pair of probabilistic Turing machines P , V , in which P is probabilistic polynomial time and V is deterministic polynomial time, P , V is called the noninteractive zero knowledge proof system for language L if the following conditions are met.

Witness indistinguishability is a weaker notion of zero knowledge, but it is sufficient to ensure the security of cryptographic protocol in some applications. It is worth mentioning that witness indistinguishability is closed under concurrent composition.

In view of the important theoretical and applied value of zero knowledge interactive proof system in the fields of computational complexity and cryptography, its inherent nature and characteristics have caused much attention, such as interactivity and the randomness of participants and auxiliary input. Oren [ 20 ] first proves that NIZK proof systems only exist for BPP languages in the plain model without any trusted set-up assumption. CRS is generated by a trusted party and is accessible to both the prover and verifier.

This model requires only the randomness of CRS, not relying on its privacy, so CRS model is more practical than interactive model. In the same year, de Santis et al. In addition, preprocessing model is stronger than CRS model because the two parties can generate CRS in the preprocessing stage. Comparing the two models, CRS model is more reasonable, general, and practical.

It is the widely accepted NIZK model now. They point out that CRS needs to be generated by a trusted third party in the single string model. However, it is difficult to find a suitable third party in practical applications. Therefore, it can be considered that the common reference string is generated by multiple parties as long as most of them are honest.

Meanwhile, they also present the first NIZK proof system in the multistring model. Later, Blum et al. However, the above proof systems are constructed based on specific mathematical problems. Feige et al. At the same time, they also introduce a hiding bit model and use witness indistinguishability to turn bounded NIZK into general NIZK proof system which allows many provers to use the same random string to prove different statements.

Lapidot and Shamir [ 27 ] give the first publicly verifiable NIZK assuming the existence of one-way permutations. Then NIZK proof systems for general NP problem can be obtained by Karp reduction, but this kind of constructions engages a very high level of complexity.

Simultaneously, he also gives noninteractive statistical zero knowledge argument of HC problem under the preprocessing model. Bellare and Yung [ 29 ] point out that the trapdoor permutation used in NIZK proof system in [ 25 ] requires additional verification and puts forward the corresponding solution. Boyar et al. Statistical zero knowledge [ 1 ] plays a significant role in both practical application and theoretical study, because it reflects the inherent characteristics of zero knowledge and does not need to be constructed under cryptographic assumptions as computational zero knowledge.

NISZK and similar notions are defined, resp. Ostrovsky [ 36 ] proves that, for any nontrivial language, the existence of SZK and NISZK proof or argument system is a sufficient condition for the existence of one-way functions. Thereafter, de Santis et al. First, they give a NIPZK proof for quadratic residue in [ 37 ] and a new method that turns noninteractive proofs into interactive proofs which can not only keep the same zero knowledge characteristics but also make the round of the converted interactive proof systems optimal.

In , de Santis et al. With the help of Boolean circuit composition theory, de Santis et al. Besides, the idea also applies to SZK. Pass et al. Additionally, for the language undecidable by nonuniform polynomial circuits, the necessary and sufficient condition of NIZK is the existence of one-way function.

The above results indicate that, for general NP language, noninteractive statistical perfect zero knowledge proof does not exist. Then, does noninteractive statistical perfect zero knowledge argument exist? Groth et al. Since its invention, researches on NIZK are mainly focused on the theoretical problems. Although it is once used to construct CCA-2 secure encryption schemes by Naor and Yung [ 43 ] and signature schemes by Bellare and Goldwasser [ 44 ], these results are just theoretical feasibility without practical applications.

One of the important reasons is that the construction of NIZK is not efficient. While, in practical applications, we instead consider certain types of problems such as the computations in the bilinear group , therefore the NIZK proof systems for general NP problems have to be reduced to NIZK proof systems for specific problems, which greatly sacrifices the efficiency.

How to construct efficient NIZK proof systems seems to be the key to promote their applications. In , Groth and Sahai [ 17 ] analyze the reasons why the past NIZK proofs are inefficient and put forward the famous GS proof framework that applies to all basic operations in bilinear group. NIZK proof system can be obtained simply and efficiently through instantiating GS proof according to different application backgrounds, which greatly simplifies the design of public key cryptographic algorithm and cryptographic protocol based on bilinear groups.

Since then, Ghadafi et al. Later, Groth [ 7 — 12 ] makes further improvements on some aspects such as the computational efficiency and length of NIZK. The relationship, comparison, and transformation between NIZK and IZK are also important research directions of zero knowledge proof systems.

At first Blum et al. It is a probabilistic polynomial time algorithm for solving the common reference string. At the same time, they point out that this result applies to the computational and statistical zero knowledge, not relying on cryptographic assumptions. From then on, Chailloux et al. But hash function is used in this transformation, so the NIZK argument can only be proved to be secure in the RO model. In , Dwork and Naor show a surprising result in [ 51 ]: there exists two-round public-coin witness indistinguishable proof system that does not use CRS.

The authors call the proof system zap. In a zap, the verifier first sends a random string to the prover; then the prover replies with a message to complete the proof.

Zaps have many applications such as the construction of concurrent zero knowledge, deniable authentication [ 51 ], and ring signature [ 52 ]. As can be seen from the definition of zap, it has an important link with NIZK. In , Groth et al. The inherent privacy and authentication properties of zero knowledge proof system make it widely used in the construction of cryptographic protocols.

Generally speaking, IZK proof system is usually used to construct multiround interactive protocol in the plain model, for example, general two-party and multiparty secure computation, and mostly for designing protocols in an abstract way, while NIZK proof is usually integrated into the construction of specific, practical cryptographic algorithm and cryptographic protocols. This raises very high demands on the construction of efficient NIZK proof systems.

At first, Blum et al. However, this paper only shows the possibility but does not give a specific construction.

Bellare and Goldwasser [ 44 ] present a new method to construct signature and message authentication protocol with the help of NIZK. And the scheme obtained is secure against adaptive chosen message attack. At the same time, this paper also gives an encryption scheme secure against adaptive chosen ciphertext attack. On the other hand, NIZK is widely used in group signatures, ring signatures, and electronic voting. NIZK is first used to construct a provably secure group signature scheme in the standard model by Bellare et al.

Thereafter Groth uses NIZK to construct a group signature with constant size as well as a completely anonymous group signature [ 6 ] scheme in the standard model. Zap is introduced to the construction of ring signature for the first time by Bender et al.

In the recent 20 years, researches on NIZK proof system and related theory have improved gradually. Recent research focuses are mainly concentrated on the application and efficiency improvement of NIZK proof system, including the following aspects.

Efficient NIZK proof and NIZK argument system that apply to specific application backgrounds: currently, the researches for NIZK efficiency are mainly concentrated on the computation in bilinear group, so it is worth deeply studying how to construct highly efficient NIZK protocol applicable to other mathematical backgrounds.

Other cryptographic tools that cooperate with the existing proof systems: recently, Abe et al. At present, these researches are just beginning, and there are still a lot of problems in the efficiency and application of these schemes. The authors declare that there is no conflict of interests regarding the publication of this paper. National Center for Biotechnology Information , U.

## An authentication protocol based on chaos and zero knowledge proof

Zero knowledge proof system which has received extensive attention since it was proposed is an important branch of cryptography and computational complexity theory. Thereinto, noninteractive zero knowledge proof system contains only one message sent by the prover to the verifier. It is widely used in the construction of various types of cryptographic protocols and cryptographic algorithms because of its good privacy, authentication, and lower interactive complexity. This paper reviews and analyzes the basic principles of noninteractive zero knowledge proof system, and summarizes the research progress achieved by noninteractive zero knowledge proof system on the following aspects: the definition and related models of noninteractive zero knowledge proof system, noninteractive zero knowledge proof system of NP problems, noninteractive statistical and perfect zero knowledge, the connection between noninteractive zero knowledge proof system, interactive zero knowledge proof system, and zap, and the specific applications of noninteractive zero knowledge proof system. This paper also points out the future research directions.

zero knowledge proof system[GMW87], opening up the possibility for a vast range of efficient than a sigma-protocol-based solution for non-algebraic statements. drew M. Odlyzko, editor, CRYPTO'86, volume of LNCS, pages –

## Verifiable Credentials Data Model 1.0

Zero knowledge proof system which has received extensive attention since it was proposed is an important branch of cryptography and computational complexity theory. Thereinto, noninteractive zero knowledge proof system contains only one message sent by the prover to the verifier. It is widely used in the construction of various types of cryptographic protocols and cryptographic algorithms because of its good privacy, authentication, and lower interactive complexity. This paper reviews and analyzes the basic principles of noninteractive zero knowledge proof system, and summarizes the research progress achieved by noninteractive zero knowledge proof system on the following aspects: the definition and related models of noninteractive zero knowledge proof system, noninteractive zero knowledge proof system of NP problems, noninteractive statistical and perfect zero knowledge, the connection between noninteractive zero knowledge proof system, interactive zero knowledge proof system, and zap, and the specific applications of noninteractive zero knowledge proof system. This paper also points out the future research directions.

Number Theory Topics Pdf This book will, by necessity, touch on a number of different areas of study, and as such is more than just a text for aspiring Electrical Engineers. Modular Arithmetic. Number theory is a vast and sprawling subject, and over the years this book has acquired many new chapters.

### A Survey of Noninteractive Zero Knowledge Proof System and Its Applications

Cryptography is one of the the most important components of the blockchain technology, which has become widely spread over the last few years. You will learn about the general concepts of a ZKP and noninteractive zero-knowledge proof, see some use cases for employing the protocol within a blockchain, and get a dive into a ZKP from the perspective of cryptography. A zero-knowledge proof is one of the most abstract and fascinating concepts in applied cryptography today. From potentially being used in nuclear disarmament to providing anonymous and secure transactions for public blockchain networks, a zero-knowledge proof is a profound example of cryptographic innovation. In cryptography, a zero-knowledge proof or zero-knowledge protocol is a method by which one party the prover can prove to another party the verifier that they know a value x , without conveying any information apart from the fact that they know the value x. The essence of a zero-knowledge proof is that it is trivial to prove that someone possesses knowledge of certain information by simply revealing it.

Port Knocking is a method for authenticating clients through a closed stance firewall, and authorising their requested actions, enabling severs to offer services to authenticated clients, without opening ports on the firewall. Advances in port knocking have resulted in an increase in complexity in design, preventing port knocking solutions from realising their potential. This paper proposes a novel port knocking solution, named Crucible, which is a secure method of authentication, with high usability and features of stealth, allowing servers and services to remain hidden and protected. The solution is forwarded as a method for protecting servers against attacks ranging from port scans, to zero-day exploitation. To act as a random oracle for both client and server, cryptographic hashes were generated through chaotic systems. Port knocking, if integrated into a security environment, can offer an additional layer of authentication for servers, furthering a defence-in-depth approach, and can conceal the presence of services. It is suited to defending against attacks directed at servers, ranging from automatic scanning, as part of attack-chain reconnaissance, to precisely targeted zero-day exploitation.

Skip to Main Content. A not-for-profit organization, IEEE is the world's largest technical professional organization dedicated to advancing technology for the benefit of humanity. Use of this web site signifies your agreement to the terms and conditions. Privacy-Preserving Traffic Management: A Blockchain and Zero-Knowledge Proof Inspired Approach Abstract: Incorporation of connected vehicle CV data into real-time traffic management systems presents a host of new challenges resulting from the current lack of data integrity and data privacy in traffic networks. Over the past few years, blockchain technologies have been inspiring extensive innovations in the transportation field. However, due to the transparency property, sensitive data stored on the blockchain would be accessible to anyone, resulting in a lack of privacy.

Examples of zero-knowledge proof systems are given for the languages These are the first zero-knowledge protocols demonstrated for languages not known that A and B share the same input tape, B's write-only communication tape is A's.

#### 1. Introduction

ГЛАВА 80 Хейл, крепко сжимая шею Сьюзан, крикнул в темноту: - Коммандер, твоя подружка у меня в руках. Я требую выпустить меня отсюда. В ответ - тишина. Его руки крепче сжали ее шею. - Я сейчас ее убью.

И я постараюсь это право обеспечить. ГЛАВА 7 Мозг Сьюзан лихорадочно работал: Энсей Танкадо написал программу, с помощью которой можно создавать шифры, не поддающиеся взлому. Она никак не могла свыкнуться с этой мыслью. - Цифровая крепость, - сказал Стратмор.

Формула называется Цифровая крепость, говорилось в заметке, и доступна для ознакомления в Интернете. Программист намеревался выставить ее на аукционе и отдать тому, кто больше всех заплатит. Далее в заметке сообщалось, что, хотя алгоритм вызвал громадный интерес в Японии, несколько американских производителей программного обеспечения, прослышавших о Цифровой крепости, считают эту информацию нелепой - чем-то вроде обещания превратить свинец в золото. Формула, утверждают они, - это мистификация, к которой не следует относиться серьезно.