We prove that for every 3-player (3-prover) game $\mathcal G$ with value less than one, whose query distribution has the support $\mathcal S = \{(1,0,0), (0,1,0), (0,0,1)\}$ of hamming weight one vectors, the value of the $n$-fold parallel repetition $\mathcal G^{\otimes n}$ decays polynomially fast to zero; that is, there is a constant $c = c(\mathcal G)>0$ such that the value of the game $\mathcal G^{\otimes n}$ is at most $n^{-c}$. Following the recent work of Girish, Holmgren, Mittal, Raz and Zhan (STOC 2022), our result is the missing piece that implies a similar bound for a much more general class of multiplayer games: For $\textbf{every}$ 3-player game $\mathcal G$ over $\textit{binary questions}$ and $\textit{arbitrary answer lengths}$, with value less than 1, there is a constant $c = c(\mathcal G)>0$ such that the value of the game $\mathcal G^{\otimes n}$ is at most $n^{-c}$. Our proof technique is new and requires many new ideas. For example, we make use of the Level-$k$ inequalities from Boolean Fourier Analysis, which, to the best of our knowledge, have not been explored in this context prior to our work.
In privacy-preserving machine learning, it is less obvious to verify correct behavior of participants because they are not supposed to reveal their inputs in cleartext to other participants. It is hence important to make federated machine learning robust against data poisoning and related attacks. While input data can be related to a distributed ledger (blockchain), a less studied input is formed by the random sampling parties perform. In this paper, we describe strategies based on zero knowledge proofs to allow parties to prove they perform sampling (and other computations) correctly. We sketch a number of alternative ways to implement our idea and provide some preliminary experimental results.
We consider the problem of efficiently inferring interventional distributions in a causal Bayesian network from a finite number of observations. Let $\mathcal{P}$ be a causal model on a set $\mathbf{V}$ of observable variables on a given causal graph $G$. For sets $\mathbf{X},\mathbf{Y}\subseteq \mathbf{V}$, and setting ${\bf x}$ to $\mathbf{X}$, let $P_{\bf x}(\mathbf{Y})$ denote the interventional distribution on $\mathbf{Y}$ with respect to an intervention ${\bf x}$ to variables ${\bf x}$. Shpitser and Pearl (AAAI 2006), building on the work of Tian and Pearl (AAAI 2001), gave an exact characterization of the class of causal graphs for which the interventional distribution $P_{\bf x}({\mathbf{Y}})$ can be uniquely determined. We give the first efficient version of the Shpitser-Pearl algorithm. In particular, under natural assumptions, we give a polynomial-time algorithm that on input a causal graph $G$ on observable variables $\mathbf{V}$, a setting ${\bf x}$ of a set $\mathbf{X} \subseteq \mathbf{V}$ of bounded size, outputs succinct descriptions of both an evaluator and a generator for a distribution $\hat{P}$ that is $\varepsilon$-close (in total variation distance) to $P_{\bf x}({\mathbf{Y}})$ where $Y=\mathbf{V}\setminus \mathbf{X}$, if $P_{\bf x}(\mathbf{Y})$ is identifiable. We also show that when $\mathbf{Y}$ is an arbitrary set, there is no efficient algorithm that outputs an evaluator of a distribution that is $\varepsilon$-close to $P_{\bf x}({\mathbf{Y}})$ unless all problems that have statistical zero-knowledge proofs, including the Graph Isomorphism problem, have efficient randomized algorithms.
Joana Lorenz, Maria Inês Silva, David Aparício, João Tiago Ascensão · 5 authors
Every year, criminals launder billions of dollars acquired from serious felonies (e.g., terrorism, drug smuggling, or human trafficking), harming countless people and economies. Cryptocurrencies, in particular, have developed as a haven for money laundering activity. Machine Learning can be used to detect these illicit patterns. However, labels are so scarce that traditional supervised algorithms are inapplicable. Here, we address money laundering detection assuming minimal access to labels. First, we show that existing state-of-the-art solutions using unsupervised anomaly detection methods are inadequate to detect the illicit patterns in a real Bitcoin transaction dataset. Then, we show that our proposed active learning solution is capable of matching the performance of a fully supervised baseline by using just 5% of the labels. This solution mimics a typical real-life situation in which a limited number of labels can be acquired through manual annotation by experts.
We define a new query measure we call quantum distinguishing complexity, denoted QD(f) for a Boolean function f. Unlike a quantum query algorithm, which must output a state close to |0> on a 0-input and a state close to |1> on a 1-input, a "quantum distinguishing algorithm" can output any state, as long as the output states for any 0-input and 1-input are distinguishable. \nUsing this measure, we establish a new relationship in query complexity: For all total functions f, Q_0(f)=O~(Q(f)^5), where Q_0(f) and Q(f) denote the zero-error and bounded-error quantum query complexity of f respectively, improving on the previously known sixth power relationship. \nWe also define a query measure based on quantum statistical zero-knowledge proofs, QSZK(f), which is at most Q(f). We show that QD(f) in fact lower bounds QSZK(f) and not just Q(f). QD(f) also upper bounds the (positive-weights) adversary bound, which yields the following relationships for all f: Q(f) >= QSZK(f) >= QD(f) = Omega(Adv(f)). This sheds some light on why the adversary bound proves suboptimal bounds for problems like Collision and Set Equality, which have low QSZK complexity. \nLastly, we show implications for lifting theorems in communication complexity. We show that a general lifting theorem for either zero-error quantum query complexity or for QSZK would imply a general lifting theorem for bounded-error quantum query complexity.
In both query and communication complexity, we give separations between the class NISZK, containing those problems with non-interactive statistical zero knowledge proof systems, and the class UPP, containing those problems with randomized algorithms with unbounded error. These results significantly improve on earlier query separations of Vereschagin [Ver95] and Aaronson [Aar12] and earlier communication complexity separations of Klauck [Kla11] and Razborov and Sherstov [RS10]. In addition, our results imply an oracle relative to which the class NISZK is not contained in PP. This answers an open question of Watrous from 2002 [Aar]. The technical core of our result is a stronger hardness amplification theorem for approximate degree, which roughly says that composing the gapped-majority function with any function of high approximate degree yields a function with high threshold degree. Using our techniques, we also give oracles relative to which the following two separations hold: perfect zero knowledge (PZK) is not contained in its complement (coPZK), and SZK (indeed, even NISZK) is not contained in PZK (indeed, even HVPZK). Along the way, we show that HVPZK is contained in PP in a relativizing manner.
We prove a number of implications of these results, which may be of independent interest outside of structural complexity. Specifically, our oracle separation implies that certain parameters of the Polarization Lemma of Sahai and Vadhan [SV03] cannot be much improved in a black-box manner. Additionally, it implies new lower bounds for property testing algorithms with error probability arbitrarily close to 1/2. Finally, our results imply that two-message protocols in the streaming interactive proofs model of Cormode et al. [CTY11] are surprisingly powerful in the sense that, with just logarithmic cost, they can compute functions outside of UPP^CC.
A Zero-Knowledge PCP (ZK-PCP) is a randomized PCP such that the view of any (perhaps cheating) efficient verifier can be efficiently simulated up to small statistical distance. Kilian, Petrank, and Tardos (STOC '97) constructed ZK-PCPs for all languages in NEXP. Ishai, Mahmoody, and Sahai (TCC '12), motivated by cryptographic applications, revisited the possibility of efficient ZK-PCPs for all of NP where the PCP is encoded as a polynomial-size circuit that given a query i returns the ith symbol of the PCP. Ishai et al showed that there is no efficient ZK-PCP for NP with a non-adaptive verifier, that prepares all of its PCP queries before seeing any answers, unless NP⊆coAM and the polynomial-time hierarchy collapses. The question of whether adaptive verification can lead to efficient ZK-PCPs for NP remained open.
In this work, we resolve this question and show that any language or promise problem with efficient ZK-PCPs must be in SZK (the class of promise problems with a statistical zero-knowledge single prover proof system). Therefore, no NP-complete problem can have an efficient ZK-PCP unless NP⊆SZK (which also implies NP⊆coAM and the polynomial-time hierarchy collapses). We prove our result by reducing any promise problem with an efficient ZK-PCP to two instances of the Conditional Entropy Approximation problem defined and studied by Vadhan (FOCS'04) which is known to be complete for the class SZK.
We prove a number of general theorems about CZK, the class of problems possessing computational zero knowledge proofs. Our results are unconditional, in contrast to most previous works on CZK which rely on the assumption that one-way functions exist. We establish several new characterizations of CZK, and use these characterizations to prove results such as: 1) Honest-verifier CZK equals general CZK. 2) Public-coin CZK equals private-coin CZK. 3) CZK is closed under union (and more generally, "monotone formula closure"). 4) CZK with imperfect completeness equals CZK with perfect completeness. 5) Any problem in CZK /spl cap/ NP can be proven in computational zero knowledge by a BPP/sup NP/ prover. 6) CZK with black-box simulators equals CZK with general, non-black-box simulators. The above equalities refer to the resulting class of problems (and do not necessarily preserve other efficiency measures such as round complexity). Our approach is to combine the conditional techniques previously used in the study of CZK with the unconditional techniques developed in the study of SZK, the class of problems possessing statistical zero knowledge proofs. To enable this combination, we prove that every problem in CZK can be decomposed into a problem in SZK together with a set of instances from which a one-way function can be constructed.
The zero knowledge properties of interactive proof systems (IPSs) are studied in the case that the verifier is a 2-way probabilistic finite state automaton (2pfa). The following results are proved: A new definition of zero knowledge is introduced. This definition captures a concept of “zero knowledge” for IPSs that are used for language recognition.
An investigation of interactive proof systems (IPSs) where the verifier is a 2-way probabilistic finite state automaton (2pfa) is initiated. In this model, it is shown: Additional results concern two other classes of verifiers: 2pfa's that halt in polynomial expected time, and 2-way probabilistic pushdown automata that halt in polynomial time. In particular, IPSs with verifiers in the latter class are as powerful as IPSs where verifiers are polynomial-time probabilistic Turing machines. In a companion paper [7], zero knowledge IPSs with 2pfa verifiers are investigated.