Concurrent Knowledge-Extraction in the Public-Key Model

Knowledge extraction is a fundamental notion, modelling machine possession of values (witnesses) in a computational complexity sense. The notion provides an essential tool for cryptographic protocol design and analysis, enabling one to argue about the internal state of protocol players without ever looking at this supposedly secret state. However, when transactions are concurrent (e.g., over the Internet) with players possessing public-keys (as is common in cryptography), assuring that entities ``know'' what they claim to know, where adversaries may be well coordinated across different transactions, turns out to be much more subtle and in need of re-examination. Here, we investigate how to formally treat knowledge possession by parties (with registered public-keys) interacting over the Internet. Stated more technically, we look into the relative power of the notion of ``concurrent knowledge-extraction'' (CKE) in the concurrent zero-knowledge (CZK) bare public-key (BPK) model.Comment: 38 pages, 4 figure

Concurrently Non-Malleable Zero Knowledge in the Authenticated Public-Key Model

We consider a type of zero-knowledge protocols that are of interest for their practical applications within networks like the Internet: efficient zero-knowledge arguments of knowledge that remain secure against concurrent man-in-the-middle attacks. In an effort to reduce the setup assumptions required for efficient zero-knowledge arguments of knowledge that remain secure against concurrent man-in-the-middle attacks, we consider a model, which we call the Authenticated Public-Key (APK) model. The APK model seems to significantly reduce the setup assumptions made by the CRS model (as no trusted party or honest execution of a centralized algorithm are required), and can be seen as a slightly stronger variation of the Bare Public-Key (BPK) model from \cite{CGGM,MR}, and a weaker variation of the registered public-key model used in \cite{BCNP}. We then define and study man-in-the-middle attacks in the APK model. Our main result is a constant-round concurrent non-malleable zero-knowledge argument of knowledge for any polynomial-time relation (associated to a language in $\mathcal{NP}$), under the (minimal) assumption of the existence of a one-way function family. Furthermore,We show time-efficient instantiations of our protocol based on known number-theoretic assumptions. We also note a negative result with respect to further reducing the setup assumptions of our protocol to those in the (unauthenticated) BPK model, by showing that concurrently non-malleable zero-knowledge arguments of knowledge in the BPK model are only possible for trivial languages

Predictable arguments of knowledge

We initiate a formal investigation on the power of predictability for argument of knowledge systems for NP. Specifically, we consider private-coin argument systems where the answer of the prover can be predicted, given the private randomness of the verifier; we call such protocols Predictable Arguments of Knowledge (PAoK). Our study encompasses a full characterization of PAoK, showing that such arguments can be made extremely laconic, with the prover sending a single bit, and assumed to have only one round (i.e., two messages) of communication without loss of generality. We additionally explore PAoK satisfying additional properties (including zero-knowledge and the possibility of re-using the same challenge across multiple executions with the prover), present several constructions of PAoK relying on different cryptographic tools, and discuss applications to cryptography

Zero-Knowledge Proofs of Proximity

Interactive proofs of proximity (IPPs) are interactive proofs in which the verifier runs in time sub-linear in the input length. Since the verifier cannot even read the entire input, following the property testing literature, we only require that the verifier reject inputs that are far from the language (and, as usual, accept inputs that are in the language). In this work, we initiate the study of zero-knowledge proofs of proximity (ZKPP). A ZKPP convinces a sub-linear time verifier that the input is close to the language (similarly to an IPP) while simultaneously guaranteeing a natural zero-knowledge property. Specifically, the verifier learns nothing beyond (1) the fact that the input is in the language, and (2) what it could additionally infer by reading a few bits of the input. Our main focus is the setting of statistical zero-knowledge where we show that the following hold unconditionally (where N denotes the input length): - Statistical ZKPPs can be sub-exponentially more efficient than property testers (or even non-interactive IPPs): We show a natural property which has a statistical ZKPP with a polylog(N) time verifier, but requires Omega(sqrt(N)) queries (and hence also runtime) for every property tester. - Statistical ZKPPs can be sub-exponentially less efficient than IPPs: We show a property which has an IPP with a polylog(N) time verifier, but cannot have a statistical ZKPP with even an N^(o(1)) time verifier. - Statistical ZKPPs for some graph-based properties such as promise versions of expansion and bipartiteness, in the bounded degree graph model, with polylog(N) time verifiers exist. Lastly, we also consider the computational setting where we show that: - Assuming the existence of one-way functions, every language computable either in (logspace uniform) NC or in SC, has a computational ZKPP with a (roughly) sqrt(N) time verifier. - Assuming the existence of collision-resistant hash functions, every language in NP has a statistical zero-knowledge argument of proximity with a polylog(N) time verifier

Quantum Proofs

Quantum information and computation provide a fascinating twist on the notion of proofs in computational complexity theory. For instance, one may consider a quantum computational analogue of the complexity class \class{NP}, known as QMA, in which a quantum state plays the role of a proof (also called a certificate or witness), and is checked by a polynomial-time quantum computation. For some problems, the fact that a quantum proof state could be a superposition over exponentially many classical states appears to offer computational advantages over classical proof strings. In the interactive proof system setting, one may consider a verifier and one or more provers that exchange and process quantum information rather than classical information during an interaction for a given input string, giving rise to quantum complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit some properties from their classical counterparts, they also possess distinct and uniquely quantum features that lead to an interesting landscape of complexity classes based on variants of this model. In this survey we provide an overview of many of the known results concerning quantum proofs, computational models based on this concept, and properties of the complexity classes they define. In particular, we discuss non-interactive proofs and the complexity class QMA, single-prover quantum interactive proof systems and the complexity class QIP, statistical zero-knowledge quantum interactive proof systems and the complexity class \class{QSZK}, and multiprover interactive proof systems and the complexity classes QMIP, QMIP*, and MIP*.Comment: Survey published by NOW publisher

A Study of Separations in Cryptography: New Results and New Models

For more than 20 years, black-box impossibility results have been used to argue the infeasibility of constructing certain cryptographic primitives (e.g., key agreement) from others (e.g., one-way functions). In this dissertation we further extend the frontier of this field by demonstrating several new impossibility results as well as a new framework for studying a more general class of constructions. Our first two results demonstrate impossibility of black-box constructions of two commonly used cryptographic primitives. In our first result we study the feasibility of black-box constructions of predicate encryption schemes from standard assumptions and demonstrate strong limitations on the types of schemes that can be constructed. In our second result we study black-box constructions of constant-round zero-knowledge proofs from one-way permutations and show that, under commonly believed complexity assumptions, no such constructions exist. A widely recognized limitation of black-box impossibility results, however, is that they say nothing about the usefulness of (known) non-black-box techniques. This state of affairs is unsatisfying as we would at least like to rule out constructions using the set of techniques we have at our disposal. With this motivation in mind, in the final result of this dissertation we propose a new framework for black-box constructions with a non-black-box flavor, specifically, those that rely on zero-knowledge proofs relative to some oracle. Our framework is powerful enough to capture a large class of known constructions, however we show that the original black-box separation of key agreement from one-way functions still holds even in this non-black-box setting that allows for zero-knowledge proofs

Zero-Knowledge Proof Systems for QMA

© 2016 IEEE. Prior work has established that all problems in NP admit classical zero-knowledge proof systems, and under reasonable hardness assumptions for quantum computations, these proof systems can be made secure against quantum attacks. We prove a result representing a further quantum generalization of this fact, which is that every problem in the complexity class QMA has a quantum zero-knowledge proof system. More specifically, assuming the existence of an unconditionally binding and quantum computationally concealing commitment scheme, we prove that every problem in the complexity class QMA has a quantum interactive proof system that is zero-knowledge with respect to efficient quantum computations. Our QMA proof system is sound against arbitrary quantum provers, but only requires an honest prover to perform polynomial-time quantum computations, provided that it holds a quantum witness for a given instance of the QMA problem under consideration