Usually, a proof of a theorem contains more knowledge than the mere fact that the theorem is true. For instance, to prove that a graph is Hamiltonian it suffices to exhibit a Hamiltonian tour in it; however, this seems to contain more knowledge than the single bit Hamiltonian/non-Hamiltonian.In this paper a computational complexity theory of the “knowledge” contained in a proof is developed. Zero-knowledge proofs are defined as those proofs that convey no additional knowledge other than the correctness of the proposition in question. Examples of zero-knowledge proof systems are given for the languages of quadratic residuosity and 'quadratic nonresiduosity. These are the first examples of zero-knowledge proofs for languages not known to be efficiently recognizable.

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The knowledge complexity of interactive proof-systems

In this paper, we define and explore proofs of retrievability (PORs). A POR scheme enables an archive or back-up service (prover) to produce a concise proof that a user (verifier) can retrieve a target file F, that is, that the archive retains and reliably transmits file data sufficient for the user to recover F in its entirety.A POR may be viewed as a kind of cryptographic proof of knowledge (POK), but one specially designed to handle a large file (or bitstring) F. We explore POR protocols here in which the communication costs, number of memory accesses for the prover, and storage requirements of the user (verifier) are small parameters essentially independent of the length of F. In addition to proposing new, practical POR constructions, we explore implementation considerations and optimizations that bear on previously explored, related schemes.In a POR, unlike a POK, neither the prover nor the verifier need actually have knowledge of F. PORs give rise to a new and unusual security definition whose formulation is another contribution of our work.We view PORs as an important tool for semi-trusted online archives. Existing cryptographic techniques help users ensure the privacy and integrity of files they retrieve. It is also natural, however, for users to want to verify that archives do not delete or modify files prior to retrieval. The goal of a POR is to accomplish these checks without users having to download the files themselves. A POR can also provide quality-of-service guarantees, i.e., show that a file is retrievable within a certain time bound.

PORs: Proofs of Retrievability for Large Files

Bitcoin is a popular cryptocurrency that records all transactions in a distributed append-only public ledger called blockchain . The security of Bitcoin heavily relies on the incentive-compatible proof-of-work (PoW) based distributed consensus protocol, which is run by the network nodes called miners . In exchange for the incentive, the miners are expected to maintain the blockchain honestly. Since its launch in 2009, Bitcoin economy has grown at an enormous rate, and it is now worth about 150 billions of dollars. This exponential growth in the market value of bitcoins motivate adversaries to exploit weaknesses for profit, and researchers to discover new vulnerabilities in the system, propose countermeasures, and predict upcoming trends. In this paper, we present a systematic survey that covers the security and privacy aspects of Bitcoin. We start by giving an overview of the Bitcoin system and its major components along with their functionality and interactions within the system. We review the existing vulnerabilities in Bitcoin and its major underlying technologies such as blockchain and PoW-based consensus protocol. These vulnerabilities lead to the execution of various security threats to the standard functionality of Bitcoin. We then investigate the feasibility and robustness of the state-of-the-art security solutions. Additionally, we discuss the current anonymity considerations in Bitcoin and the privacy-related threats to Bitcoin users along with the analysis of the existing privacy-preserving solutions. Finally, we summarize the critical open challenges, and we suggest directions for future research towards provisioning stringent security and privacy solutions for Bitcoin.

A Survey on Security and Privacy Issues of Bitcoin

In recent years, the rapid urbanization of world’s population causes many economic, social, and environmental problems, which affect people’s living conditions and quality of life significantly. The concept of “smart city” brings opportunities to solve these urban problems. The objectives of smart cities are to make the best use of public resources, provide high-quality services to the citizens, and improve the people’s quality of life. Information and communication technology plays an important role in the implementation of smart cities. Blockchain as an emerging technology has many good features, such as trust-free, transparency, pseudonymity, democracy, automation, decentralization, and security. These features of blockchain are helpful to improve smart city services and promote the development of smart cities. In this paper, we provide a comprehensive survey on the literature involving blockchain technology applied to smart cities. First, the related works and background knowledge are introduced. Then, we review how blockchain technology is applied in the realm of smart cities, from the perspectives of smart citizen, smart healthcare, smart grid, smart transportation, supply chain management, and others. Finally, some challenges and broader perspectives are discussed.

A Survey of Blockchain Technology Applied to Smart Cities: Research Issues and Challenges

We study the problem of building a verifiable delay function (VDF). A \(\text {VDF}\)requires a specified number of sequential steps to evaluate, yet produces a unique output that can be efficiently and publicly verified. \(\text {VDF}\)s have many applications in decentralized systems, including public randomness beacons, leader election in consensus protocols, and proofs of replication. We formalize the requirements for \(\text {VDF}\)s and present new candidate constructions that are the first to achieve an exponential gap between evaluation and verification time.

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Verifiable Delay Functions

Functional encryption (FE) is an extremely powerful cryptographic mechanism that lets an authorized entity compute on encrypted data, and learn the results in the clear. However, all current cryptographic instantiations for general FE are too impractical to be implemented. We construct IRON, a provably secure, and practical FE system using Intel's recent Software Guard Extensions (SGX). We show that IRON can be applied to complex functionalities, and even for simple functions, outperforms the best known cryptographic schemes. We argue security by modeling FE in the context of hardware elements, and prove that IRON satisfies the security model.

IRON: Functional Encryption using Intel SGX

We construct a new polynomial commitment scheme for univariate and multivariate polynomials over finite fields, with logarithmic size evaluation proofs and verification time, measured in the number of coefficients of the polynomial. The underlying technique is a Diophantine Argument of Knowledge (DARK), leveraging integer representations of polynomials and groups of unknown order. Security is shown from the strong RSA and the adaptive root assumptions. Moreover, the scheme does not require a trusted setup if instantiated with class groups. We apply this new cryptographic compiler to a restricted class of algebraic linear IOPs, which we call Polynomial IOPs, to obtain doubly-efficient public-coin interactive arguments of knowledge for any NP relation with succinct communication. With linear preprocessing, the online verifier’s work is logarithmic in the circuit complexity of the relation.

Transparent SNARKs from DARK Compilers.

A verifiable delay function (VDF) is an important tool used for adding delay in decentralized applications. This short note briefly surveys and compares two recent beautiful Verifiable Delay Functions (VDFs), one due to Pietrzak and the other due to Wesolowski. We also provide a new computational proof of security for one of them, and compare the complexity assumptions needed for both schemes. 1 What is a Verifiable Delay Function? A verifiable delay function (VDF) [12, 3] is a function f : X → Y that takes a prescribed time to compute, even on a parallel computer. However once computed, the output can be quickly verified by anyone. Moreover, every input x ∈ X must have a unique valid output y ∈ Y. In more detail, a VDF that implements a function X → Y is a tuple of three algorithms: • Setup(λ, T ) → pp is a randomized algorithm that takes a security parameter λ and a time bound T , and outputs public parameters pp, • Eval(pp, x)→ (y, π) takes an input x ∈ X and outputs a y ∈ Y and a proof π. • Verify(pp, x, y, π) → {accept, reject} outputs accept if y is the correct evaluation of the VDF on input x. If (y, π) ← F (pp, x) then Verify(pp, x, y, π) = accept, for all x ∈ X and pp output by Setup(λ, T ). A VDF must satisfy three properties. We state these properties informally and refer to [3] for a complete definition: • -evaluation time: algorithm Eval(pp, x) runs in time at most (1 + )T , for all x ∈ X and all pp output by Setup(λ, T ). We will explain how to measure run time in the next section. • Sequentiality: a parallel algorithm A, using at most poly(λ) processors, that runs in time less than T cannot compute the function. Specifically, for a random x ∈ X and pp output by Setup(λ, T ), if (y, π)← Eval(pp, x) then Pr [ A(pp, x) = y ] is negligible. • Uniqueness: for an input x ∈ X , exactly one y ∈ Y will be accepted by Verify. Specifically, let A be an efficient algorithm that given pp as input, outputs (x, y, π) such that Verify(pp, x, y, π) = accept. Then Pr[Eval(pp, x) 6= y] is negligible. VDFs have many applications. They are useful for constructing a verifiable randomness beacon, and they provide a “proof of elapsed time” for certain blockchain designs [7]. We refer to [3, Sec. 2] for a survey of their applications.

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Ben Fisch

Papers

Verifiable Delay Functions

IRON: Functional Encryption using Intel SGX

Transparent SNARKs from DARK Compilers.

A Survey of Two Verifiable Delay Functions.

Transparent SNARKs from DARK Compilers