Non-Interactive Proofs: What Assumptions Are Sufficient?

A non-Interactive proof system allows a prover to convince a verifier that a statement is true by sending a single round of messages. In this thesis, we study under what assumptions can we build non-interactive proof systems with succinct verification and zero-knowledge. We obtain the following results. - Succinct Arguments: We construct the first non-interactive succinct arguments (SNARGs) for P from standard assumptions. Our construction is based on the polynomial hardness of Learning with Errors (LWE). - Zero-Knowledge: We build the first non-interactive zero-knowledge proof systems (NIZKs) for NP from sub-exponential Decisional Diffie-Hellman (DDH) assumption in the standard groups, without use of groups with pairings. To obtain our results, we build SNARGs for batch-NP from LWE and correlation intractable hash functions for TC^0 from sub-exponential DDH assumption, respectively, which may be of independent interest

Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation

We present an efficient proof system for Multipoint Arithmetic Circuit Evaluation: for every arithmetic circuit $C(x_1,\ldots,x_n)$ of size $s$ and degree $d$ over a field ${\mathbb F}$, and any inputs $a_1,\ldots,a_K \in {\mathbb F}^n$, $\bullet$ the Prover sends the Verifier the values $C(a_1), \ldots, C(a_K) \in {\mathbb F}$ and a proof of $\tilde{O}(K \cdot d)$ length, and $\bullet$ the Verifier tosses $\textrm{poly}(\log(dK|{\mathbb F}|/\varepsilon))$ coins and can check the proof in about $\tilde{O}(K \cdot(n + d) + s)$ time, with probability of error less than $\varepsilon$. For small degree $d$, this "Merlin-Arthur" proof system (a.k.a. MA-proof system) runs in nearly-linear time, and has many applications. For example, we obtain MA-proof systems that run in $c^{n}$ time (for various $c < 2$) for the Permanent, $\#$Circuit-SAT for all sublinear-depth circuits, counting Hamiltonian cycles, and infeasibility of $0$-$1$ linear programs. In general, the value of any polynomial in Valiant's class ${\sf VP}$ can be certified faster than "exhaustive summation" over all possible assignments. These results strongly refute a Merlin-Arthur Strong ETH and Arthur-Merlin Strong ETH posed by Russell Impagliazzo and others. We also give a three-round (AMA) proof system for quantified Boolean formulas running in $2^{2n/3+o(n)}$ time, nearly-linear time MA-proof systems for counting orthogonal vectors in a collection and finding Closest Pairs in the Hamming metric, and a MA-proof system running in $n^{k/2+O(1)}$-time for counting $k$-cliques in graphs. We point to some potential future directions for refuting the Nondeterministic Strong ETH.Comment: 17 page

3-Message Zero Knowledge Against Human Ignorance

The notion of Zero Knowledge has driven the field of cryptography since its conception over thirty years ago. It is well established that two-message zero-knowledge protocols for NP do not exist, and that four-message zero-knowledge arguments exist under the minimal assumption of one-way functions. Resolving the precise round complexity of zero-knowledge has been an outstanding open problem for far too long. In this work, we present a three-message zero-knowledge argument system with soundness against uniform polynomial-time cheating provers. The main component in our construction is the recent delegation protocol for RAM computations (Kalai and Paneth, TCC 2016B and Brakerski, Holmgren and Kalai, ePrint 2016). Concretely, we rely on a three-message variant of their protocol based on a key-less collision-resistant hash functions secure against uniform adversaries as well as other standard primitives. More generally, beyond uniform provers, our protocol provides a natural and meaningful security guarantee against real-world adversaries, which we formalize following Rogaway’s “human-ignorance” approach (VIETCRYPT 2006): in a nutshell, we give an explicit uniform reduction from any adversary breaking the soundness of our protocol to finding collisions in the underlying hash function.National Science Foundation (U.S.) (Award CNS-1350619)National Science Foundation (U.S.) (Award CNS-1413964

Witness Indistinguishability for any Single-Round Argument with Applications to Access Control

Consider an access policy for some resource which only allows access to users of the system who own a certain set of attributes. Specifically, we consider the case where such an access structure is defined by some monotone function $f:\{0,1\}^N \rightarrow \{0,1\}$, belonging to some class of function $F$ (e.g.\ conjunctions, space bounded computation), where $N$ is the number of possible attributes. In this work we show that any succinct single-round delegation scheme for the function class $F$ can be converted into a succinct single-round private access control protocol. That is, a verifier can be convinced that an approved user (i.e.\ one which holds an approved set of attributes) is accessing the system, without learning any additional information about the user or the set of attributes. As a main tool of independent interest, we show that assuming a quasi-polynomially secure two-message oblivious transfer scheme with statistical sender privacy (which can be based on quasi-polynomial hardness of the DDH, QR, DCR or LWE assumptions), we can convert any single-round protocol into a witness indistinguishable one, with similar communication complexity

Non-Interactive Delegation for Low-Space Non-Deterministic Computation

We construct a delegation scheme for verifying non-deterministic computations, with complexity proportional only to the non-deterministic space of the computation. Specifically, letting $n$ denote the input length, we construct a delegation scheme for any language verifiable in non-deterministic time and space $(T (n); S(n))$ with communication complexity $poly(S(n))$, verifier runtime $n polylog(T (n)) + poly(S(n))$, and prover runtime $poly(T (n))$. Our scheme consists of only two messages and has adaptive soundness, assuming the existence of a sub-exponentially secure private information retrieval (PIR) scheme, which can be instantiated under standard (albeit, sub-exponential) cryptographic assumptions, such as the sub-exponential LWE assumption. Specifically, the verifier publishes a (short) public key ahead of time, and this key can be used by any prover to non-interactively prove the correctness of any adaptively chosen non-deterministic computation. Such a scheme is referred to as a noninteractive delegation scheme. Our scheme is privately verifiable, where the verifier needs the corresponding secret key in order to verify proofs. Prior to our work, such results were known only in the Random Oracle Model, or under knowledge assumptions. Our results yield succinct non-interactive arguments based on subexponential LWE, for many natural languages believed to be outside of P

Correlation Intractability and SNARGs from Sub-exponential DDH

We provide the first constructions of SNARGs for Batch-NP and P based solely on the sub-exponential Decisional Diffie Hellman (DDH) assumption. Our schemes achieve poly-logarithmic proof sizes. Central to our results and of independent interest is a new construction of correlation-intractable hash functions for ``small input\u27\u27 product relations verifiable in $\mathsf{TC}^0$, based on sub-exponential DDH

Non-Interactive RAM and Batch NP Delegation from any PIR

We present an adaptive and non-interactive protocol for verifying arbitrary efficient computations in fixed polynomial time. Our protocol is computationally sound and can be based on any computational PIR scheme, which in turn can be based on standard polynomial-time cryptographic assumptions (e.g. the worst case hardness of polynomial-factor approximation of short-vector lattice problems). In our protocol, the prover and the verifier do not need to interact at all: The verifier sets up a public key ahead of time, and this key can be used by any prover to prove arbitrary statements in a completely adaptive manner. Verification is done using a secret verification key, and soundness relies on this key not being known to the prover. Our protocol further allows to prove statements about computations of arbitrary RAM machines. Previous works either relied on knowledge assumptions, or could only offer non-adaptive two-message protocols (where the first message could not be re-used), and required either obfuscation-based assumptions or super-polynomial hardness assumptions. We show that our techniques can also be applied to construct a new type of (non-adaptive) 2-message delegation protocols for batch NP statements. Specifically, we can simultaneously prove the membership of multiple instances in a given NP language, with communication complexity proportional to the length of a single witness

Verifiable Private Information Retrieval

A computational PIR scheme allows a client to privately query a database hosted on a single server without downloading the entire database. We introduce the notion of verifiable PIR (vPIR) where the server can convince the client that the database satisfies certain properties without additional rounds and while keeping the communication sub-linear. For example, the server can prove that the number of rows in the database that satisfy a predicate $P$ is exactly $n$. We define security by modeling vPIR as an ideal functionality and following the real-ideal paradigm. Starting from a standard PIR scheme, we construct a vPIR scheme for any database property that can be verified by a machine that reads the database once and maintains a bounded size state between rows. We also construct vPIR with public verification based on LWE or on DLIN. The main technical hurdle is to demonstrate a simulator that extracts a long input from an adversary that sends a single short message. Our vPIR constructions are based on the notion of batch argument for NP. As contribution of independent interest, we show that batch arguments are equivalent to quasi-arguments---a relaxation of SNARKs which is known to imply succinct argument for various sub-classes of NP