We consider the problem of optimal transport between two high-dimensional distributions μ,ν in ℝⁿ from a new algorithmic perspective, in which we are given a sample x ∼ μ and we have to find a close y ∼ ν while running in poly(n) time, where n is the size/dimension of x,y. In other words, we are interested in making the running time bounded in dimension of the spaces rather than bounded in the total size of the representations of the two distributions. Our main result is a general algorithmic transport result between any product distribution μ and an arbitrary distribution ν of total cost Δ + δ under 𝓁_p^p cost; here Δ is the cost of the so-called Knothe–Rosenblatt transport from μ to ν, while δ is a computational error that goes to zero for larger running time in the transport algorithm. For this result, we need ν to be "sequentially samplable" with a "bounded average sampling cost" which is a novel but natural notion of independent interest. In addition, we prove the following. - We prove an algorithmic version of the celebrated Talagrand’s inequality for transporting the standard Gaussian distribution Φⁿ to an arbitrary ν under the Euclidean-squared cost. When ν is Φⁿ conditioned on a set S of measure ε, we show how to implement the needed sequential sampler for ν in expected time poly(n/ε), using membership oracle access to S. Hence, we obtain an algorithmic transport that maps Φⁿ to Φⁿ|S in time poly(n/ε) and expected Euclidean-squared distance O(log 1/ε), which is optimal for a general set S of measure ε. - As corollary, we find the first computational concentration (Etesami et al. SODA 2020) result for the Gaussian measure under the Euclidean distance with a dimension-independent transportation cost, resolving a question of Etesami et al. More precisely, for any set S of Gaussian measure ε, we map most of Φⁿ samples to S with Euclidean distance O(√{log 1/ε}) in time poly(n/ε).

How can a model owner prove they trained their model according to the correct specification? More importantly, how can they do so while preserving the privacy of the underlying dataset and the final model? We study this problem and formulate the notion of zero-knowledge proof of training (zkPoT), which formalizes rigorous security guarantees that should be achieved by a privacy-preserving proof of training. While it is theoretically possible to design zkPoT for any model using generic zero-knowledge proof systems, this approach results in extremely unpractical proof generation times. Towards designing a practical solution, we propose the idea of combining techniques from MPC-in-the-head and zkSNARKs literature to strike an appropriate trade-off between proof size and proof computation time. We instantiate this idea and propose a concretely efficient, novel zkPoT protocol for logistic regression.

Overparameterized models with millions of parameters have been hugely successful. In this work, we ask: can the need for large models be, at least in part, due to the \emph{computational} limitations of the learner? Additionally, we ask, is this situation exacerbated for \emph{robust} learning? We show that this indeed could be the case. We show learning tasks for which computationally bounded learners need \emph{significantly more} model parameters than what information-theoretic learners need. Furthermore, we show that even more model parameters could be necessary for robust learning. In particular, for computationally bounded learners, we extend the recent result of Bubeck and Sellke [NeurIPS'2021] which shows that robust models might need more parameters, to the computational regime and show that bounded learners could provably need an even larger number of parameters. Then, we address the following related question: can we hope to remedy the situation for robust computationally bounded learning by restricting \emph{adversaries} to also be computationally bounded for sake of obtaining models with fewer parameters? Here again, we show that this could be possible. Specifically, building on the work of Garg, Jha, Mahloujifar, and Mahmoody [ALT'2020], we demonstrate a learning task that can be learned efficiently and robustly against a computationally bounded attacker, while to be robust against an information-theoretic attacker requires the learner to utilize significantly more parameters.

Consider the task of learning a hypothesis class $\mathcal{H}$ in the presence of an adversary that can replace up to an $\eta$ fraction of the examples in the training set with arbitrary adversarial examples. The adversary aims to fail the learner on a particular target test point $x$ which is \emph{known} to the adversary but not to the learner. In this work we aim to characterize the smallest achievable error $\epsilon=\epsilon(\eta)$ by the learner in the presence of such an adversary in both realizable and agnostic settings. We fully achieve this in the realizable setting, proving that $\epsilon=\Theta(\mathtt{VC}(\mathcal{H})\cdot \eta)$, where $\mathtt{VC}(\mathcal{H})$ is the VC dimension of $\mathcal{H}$. Remarkably, we show that the upper bound can be attained by a deterministic learner. In the agnostic setting we reveal a more elaborate landscape: we devise a deterministic learner with a multiplicative regret guarantee of $\epsilon \leq C\cdot\mathtt{OPT} + O(\mathtt{VC}(\mathcal{H})\cdot \eta)$, where $C > 1$ is a universal numerical constant. We complement this by showing that for any deterministic learner there is an attack which worsens its error to at least $2\cdot \mathtt{OPT}$. This implies that a multiplicative deterioration in the regret is unavoidable in this case. Finally, the algorithms we develop for achieving the optimal rates are inherently improper. Nevertheless, we show that for a variety of natural concept classes, such as linear classifiers, it is possible to retain the dependence $\epsilon=\Theta_{\mathcal{H}}(\eta)$ by a proper algorithm in the realizable setting. Here $\Theta_{\mathcal{H}}$ conceals a polynomial dependence on $\mathtt{VC}(\mathcal{H})$.

Privacy attacks on machine learning models aim to identify the data that is used to train such models. Such attacks, traditionally, are studied on static models that are trained once and are accessible by the adversary. Motivated to meet new legal requirements, many machine learning methods are recently extended to support machine unlearning, i.e., updating models as if certain examples are removed from their training sets, and meet new legal requirements. However, privacy attacks could potentially become more devastating in this new setting, since an attacker could now access both the original model before deletion and the new model after the deletion. In fact, the very act of deletion might make the deleted record more vulnerable to privacy attacks. Inspired by cryptographic definitions and the differential privacy framework, we formally study privacy implications of machine unlearning. We formalize (various forms of) deletion inference and deletion reconstruction attacks, in which the adversary aims to either identify which record is deleted or to reconstruct (perhaps part of) the deleted records. We then present successful deletion inference and reconstruction attacks for a variety of machine learning models and tasks such as classification, regression, and language models. Finally, we show that our attacks would provably be precluded if the schemes satisfy (variants of) deletion compliance (Garg, Goldwasser, and Vasudevan, Eurocrypt’20).

Black-box separations have been successfully used to identify the limits of a powerful set of tools in cryptography, namely those of black-box reductions. They allow proving that a large set of techniques are not capable of basing one primitive 𝒫 on another 𝒬. Such separations, however, do not say anything about the power of the combination of primitives 𝒬₁,𝒬₂ for constructing 𝒫, even if 𝒫 cannot be based on 𝒬₁ or 𝒬₂ alone. By introducing and formalizing the notion of black-box uselessness, we develop a framework that allows us to make such conclusions. At an informal level, we call primitive 𝒬 black-box useless (BBU) for 𝒫 if 𝒬 cannot help constructing 𝒫 in a black-box way, even in the presence of another primitive 𝒵. This is formalized by saying that 𝒬 is BBU for 𝒫 if for any auxiliary primitive 𝒵, whenever there exists a black-box construction of 𝒫 from (𝒬,𝒵), then there must already also exist a black-box construction of 𝒫 from 𝒵 alone. We also formalize various other notions of black-box uselessness, and consider in particular the setting of efficient black-box constructions when the number of queries to 𝒬 is below a threshold. Impagliazzo and Rudich (STOC'89) initiated the study of black-box separations by separating key agreement from one-way functions. We prove a number of initial results in this direction, which indicate that one-way functions are perhaps also black-box useless for key agreement. In particular, we show that OWFs are black-box useless in any construction of key agreement in either of the following settings: (1) the key agreement has perfect correctness and one of the parties calls the OWF a constant number of times; (2) the key agreement consists of a single round of interaction (as in Merkle-type protocols). We conjecture that OWFs are indeed black-box useless for general key agreement. We also show that certain techniques for proving black-box separations can be lifted to the uselessness regime. In particular, we show that the lower bounds of Canetti, Kalai, and Paneth (TCC'15) as well as Garg, Mahmoody, and Mohammed (Crypto'17 & TCC'17) for assumptions behind indistinguishability obfuscation (IO) can be extended to derive black-box uselessness of a variety of primitives for obtaining (approximately correct) IO. These results follow the so-called "compiling out" technique, which we prove to imply black-box uselessness. Eventually, we study the complementary landscape of black-box uselessness, namely black-box helpfulness. We put forth the conjecture that one-way functions are black-box helpful for building collision-resistant hash functions. We define two natural relaxations of this conjecture, and prove that both of these conjectures are implied by a natural conjecture regarding random permutations equipped with a collision finder oracle, as defined by Simon (Eurocrypt'98). This conjecture may also be of interest in other contexts, such as amplification of hardness.

Poisoning attacks have emerged as a significant security threat to machine learning algorithms. It has been demonstrated that adversaries who make small changes to the training set, such as adding specially crafted data points, can hurt the performance of the output model. Most of these attacks require the full knowledge of training data. This leaves open the possibility of achieving the same attack results using poisoning attacks that do not have the full knowledge of the clean training set.In this work, we initiate a theoretical study of the problem above. Specifically, for the case of feature selection with LASSO, we show that \emph{full information} adversaries (that craft poisoning examples based on the rest of the training data) are provably much more devastating compared to the optimal attacker that is \emph{oblivious} to the training set yet has access to the distribution of the data. Our separation result shows that the two settings of data-aware and data-oblivious are fundamentally different and we cannot hope to achieve the same attack or defense results in these scenarios.

A private machine learning algorithm hides as much as possible about its training data while still preserving accuracy. In this work, we study whether a non-private learning algorithm can be made private by relying on an instance-encoding mechanism that modifies the training inputs before feeding them to a normal learner. We formalize both the notion of instance encoding and its privacy by providing two attack models. We first prove impossibility results for achieving a (stronger) model. Next, we demonstrate practical attacks in the second (weaker) attack model on InstaHide, a recent proposal by Huang, Song, Li and Arora [ICML’20] that aims to use instance encoding for privacy.

In this paper, we study PAC learnability and certification under instance-targeted poisoning attacks, where the adversary may change a fraction of the training set with the goal of fooling the learner at a specific target instance. Our first contribution is to formalize the problem in various settings, and explicitly discussing subtle aspects such as learner’s randomness and whether (or not) adversary’s attack can depend on it. We show that when the budget of the adversary scales sublinearly with the sample complexity, PAC learnability and certification are achievable. In contrast, when the adversary’s budget grows linearly with the sample complexity, the adversary can potentially drive up the expected 0-1 loss to one. We also study distribution-specific PAC learning in the same attack model and show that proper learning with certification is possible for learning half spaces under natural distributions. Finally, we empirically study the robustness of K nearest neighbour, logistic regression, multi-layer perceptron, and convolutional neural network on real data sets against targeted-poisoning attacks. Our experimental results show that many models, especially state-of-the-art neural networks, are indeed vulnerable to these strong attacks. Interestingly, we observe that methods with high standard accuracy might be more vulnerable to instance-targeted poisoning attacks.

Over recent years, devising classification algorithms that are robust to adversarial perturbations has emerged as a challenging problem. In particular, deep neural nets (DNNs) seem to be susceptible to small imperceptible changes over test instances. However, the line of work in provable robustness, so far, has been focused on information theoretic robustness, ruling out even the existence of any adversarial examples. In this work, we study whether there is a hope to benefit from algorithmic nature of an attacker that searches for adversarial examples, and ask whether there is any learning task for which it is possible to design classifiers that are only robust against polynomial-time adversaries. Indeed, numerous cryptographic tasks (e.g. encryption of long messages) can only be secure against computationally bounded adversaries, and are indeed impossible for computationally unbounded attackers. Thus, it is natural to ask if the same strategy could help robust learning. We show that computational limitation of attackers can indeed be useful in robust learning by demonstrating the possibility of a classifier for some learning task for which computational and information theoretic adversaries of bounded perturbations have very different power. Namely, while computationally unbounded adversaries can attack successfully and find adversarial examples with small perturbation, polynomial time adversaries are unable to do so unless they can break standard cryptographic hardness assumptions. Our results, therefore, indicate that perhaps a similar approach to cryptography (relying on computational hardness) holds promise for achieving computationally robust machine learning. On the reverse directions, we also show that the existence of such learning task in which computational robustness beats information theoretic robustness requires computational hardness by implying (average-case) hardness of NP.

Boneh, Bonneau, Bünz, and Fisch (CRYPTO 2018) recently introduced the notion of a verifiable delay function (VDF). VDFs are functions that take a long sequential time T to compute, but whose outputs y := Eval(x) can be efficiently verified (possibly given a proof π) in time t ≪ T (e.g., t = poly(λ, log T) where λ is the security parameter). The first security requirement on a VDF, called uniqueness, is that no polynomial-time algorithm can find a convincing proof π' that verifies for an input x and a different output y' ≠ y. The second security requirement, called sequentiality, is that no polynomial-time algorithm running in time σ < T for some parameter σ (e.g., σ = T^{1/10}) can compute y, even with poly(T,λ) many parallel processors. Starting from the work of Boneh et al., there are now multiple constructions of VDFs from various algebraic assumptions. In this work, we study whether VDFs can be constructed from ideal hash functions in a black-box way, as modeled in the random oracle model (ROM). In the ROM, we measure the running time by the number of oracle queries and the sequentiality by the number of rounds of oracle queries. We rule out two classes of constructions of VDFs in the ROM: - We show that VDFs satisfying perfect uniqueness (i.e., VDFs where no different convincing solution y' ≠ y exists) cannot be constructed in the ROM. More formally, we give an attacker that finds the solution y in ≈ t rounds of queries, asking only poly(T) queries in total. - We also rule out tight verifiable delay functions in the ROM. Tight verifiable delay functions, recently studied by Döttling, Garg, Malavolta, and Vasudevan (ePrint Report 2019), require sequentiality for σ ≈ T-T^ρ for some constant 0 < ρ < 1. More generally, our lower bound also applies to proofs of sequential work (i.e., VDFs without the uniqueness property), even in the private verification setting, and sequentiality σ > T-(T)/(2t) for a concrete verification time t.

In this work, we study probably approximately correct (PAC) learning under general perturbation-based adversarial attacks. In the most basic setting, referred to as an evasion attack, the adversary's goal is to misclassify an honestly sampled point x by adversarially perturbing it into x̃, i.e., h(x̃) = ≠ c(x̃), where c is the ground truth concept and h is the learned hypothesis. The only limitation on the adversary is that x̃ is not “too far” from x, controlled by a metric measure. We first prove that for many theoretically natural input spaces of high dimension n (e.g., isotropic Gaussian in dimension n under ℓ2perturbations), if the adversary is allowed to apply up to a sublinear amount of perturbations in the expected norm, PAC learning requires sample complexity that is exponential in the data dimension n. We then formalize hybrid attacks in which the evasion attack is preceded by a poisoning attack in which a poisoning phase is followed by specific evasion attacks. Special forms of hybrid attacks include so-called “backdoor attacks” but here we focus on the general setting in which adversary's evasion attack is only controlled by a pre-specified amount of perturbation based on data dimension and aim to misclassifying the perturbed instances. We show that PAC learning is sometimes impossible under such hybrid attacks, while it is possible without the attack (e.g., due to the bounded VC dimension).

Product measures of dimension n are known to be “concentrated” under Hamming distance. More precisely, for any set S in the product space of probability Pr[S] ≥ ε, a random point in the space, with probability 1 – δ, has a neighbor in S that is different from the original point in only coordinates (and this is optimal). In this work, we obtain the tight computational (algorithmic) version of this result, showing how given a random point and access to an S-membership query oracle, we can find such a close point of Hamming distance in time poly(n, 1/ε, 1/δ). This resolves an open question of [MM19] who proved a weaker result (that works only for ). As corollaries, we obtain polynomial-time poisoning and (in certain settings) evasion attacks against learning algorithms when the original vulnerabilities have any cryptographically non-negligible probability. We call our algorithm MUCIO (short for “MUltiplicative Conditional Influence Optimizer”) since proceeding through the coordinates of the product space, it decides to change each coordinate of the given point based on a multiplicative version of the influence of a variable, where the influence is computed conditioned on the value of all previously updated coordinates. MUCIO is an online algorithm in that it decides on the i'th coordinate of the output given only the first i coordinates of the input. It also does not make any convexity assumption about the set S. Motivated by obtaining algorithmic variants of measure concentration in other metric probability spaces, we define a new notion of algorithmic reduction between computational concentration of measure in different probability metric spaces. This notion, whose definition has some subtlety, requires two (inverse) algorithmic mappings one of which is an algorithmic Lipschitz mapping and the other one is an algorithmic coupling connecting the two distributions. As an application, we apply this notion of reduction to obtain computational concentration of measure for high-dimensional Gaussian distributions under the ℓ1 distance. We further prove several extensions to the results above as follows. (1) Generalizing in another dimension, our computational concentration result is also true when the Hamming distance is weighted. (2) As measure concentration is usually proved for concentration around mean, we show how to use our results above to obtain algorithmic concentration for that setting as well. In particular, we prove a computational variant of McDiarmid's inequality, when properly defined. (3) Our result generalizes to discrete random processes (instead of just product distributions), and this generalization leads to new tampering algorithms for collective coin tossing protocols. (4) Finally, we prove exponential lower bounds on the average running time of non-adaptive query algorithms for proving computational concentration for the case of product spaces. Perhaps surprisingly, such lower bound shows any efficient algorithm must query about S-membership of points that are not close to the original point even though we are only interested in finding a close point in S.

Many modern machine learning classifiers are shown to be vulnerable to adversarial perturbations of the instances. Despite a massive amount of work focusing on making classifiers robust, the task seems quite challenging. In this work, through a theoretical study, we investigate the adversarial risk and robustness of classifiers and draw a connection to the well-known phenomenon of “concentration of measure” in metric measure spaces. We show that if the metric probability space of the test instance is concentrated, any classifier with some initial constant error is inherently vulnerable to adversarial perturbations.One class of concentrated metric probability spaces are the so-called Lévy families that include many natural distributions. In this special case, our attacks only need to perturb the test instance by at most O(√n) to make it misclassified, where n is the data dimension. Using our general result about Lévy instance spaces, we first recover as special case some of the previously proved results about the existence of adversarial examples. However, many more Lévy families are known (e.g., product distribution under the Hamming distance) for which we immediately obtain new attacks that find adversarial examples of distance O(√n).Finally, we show that concentration of measure for product spaces implies the existence of forms of “poisoning” attacks in which the adversary tampers with the training data with the goal of degrading the classifier. In particular, we show that for any learning algorithm that uses m training examples, there is an adversary who can increase the probability of any “bad property” (e.g., failing on a particular test instance) that initially happens with non-negligible probability to ≈ 1 by substituting only Õe(√m) of the examples with other (still correctly labeled) examples.

Making learners robust to adversarial perturbation at test time (i.e., evasion attacks finding adversarial examples) or training time (i.e., data poisoning attacks) has emerged as a challenging task. It is known that in some cases \emph{sublinear} perturbations in the training phase or the testing phase can drastically decrease the quality of the predictions. These negative results, however, only prove the \emph{existence} of such successful adversarial perturbations. A natural question for these settings is whether or not we can make classifiers \emph{computationally} robust to \emph{polynomial-time} attacks. In this work, we prove some barriers against achieving such envisioned computational robustness for evasion attacks (for specific metric probability spaces) as well as poisoning attacks. In particular, we show that if the test instances come from a product distribution (e.g., uniform over $\{0,1\}^n$ or $[0,1]^n$, or isotropic $n$-variate Gaussian) and that there is an initial constant error, then there exists a \emph{polynomial-time} attack that finds adversarial examples of Hamming distance $O(\sqrt n)$. For poisoning attacks, we prove that for any deterministic learning algorithm with sample complexity $m$ and any efficiently computable “predicate” defining some “bad” property $B$ for the produced hypothesis (e.g., failing on a particular test) that happens with an initial constant probability, there exist a \emph{polynomial-time} online poisoning attack that replaces $O (\sqrt m)$ of the training examples with other correctly labeled examples and increases the probability of the bad event $B$ to $\approx 1$. Both of our poisoning and evasion attacks are \emph{black-box} in how they access their corresponding components of the system (i.e., the hypothesis, the concept, and the learning algorithm).

Many recent works have shown that adversarial examples that fool classifiers can be found by minimally perturbing a normal input. Recent theoretical results, starting with Gilmer et al. (2018b), show that if the inputs are drawn from a concentrated metric probability space, then adversarial examples with small perturbation are inevitable. A concentrated space has the property that any subset with Ω(1) (e.g.,1/100) measure, according to the imposed distribution, has small distance to almost all (e.g., 99/100) of the points in the space. It is not clear, however, whether these theoretical results apply to actual distributions such as images. This paper presents a method for empirically measuring and bounding the concentration of a concrete dataset which is proven to converge to the actual concentration. We use it to empirically estimate the intrinsic robustness to and L2 and Linfinity perturbations of several image classification benchmarks. Code for our experiments is available at https://github.com/xiaozhanguva/Measure-Concentration.

Recently, Mahloujifar and Mahmoody (TCC’17) studied attacks against learning algorithms using a special case of Valiant’s malicious noise, called $p$-tampering, in which the adversary gets to change any training example with independent probability $p$ but is limited to only choose ‘adversarial’ examples with correct labels. They obtained $p$-tampering attacks that increase the error probability in the so called ‘targeted’ poisoning model in which the adversary’s goal is to increase the loss of the trained hypothesis over a particular test example. At the heart of their attack was an efficient algorithm to bias the average output of any bounded real-valued function through $p$-tampering. In this work, we present new biasing attacks for biasing the average output of bounded real-valued functions. Our new biasing attacks achieve in \emph{polynomial-time} the the best bias achieved by MM16 through an \emph{exponential} time $p$-tampering attack. Our improved biasing attacks, directly imply improved $p$-tampering attacks against learners in the targeted poisoning model. As a bonus, our attacks come with considerably simpler analysis compared to previous attacks. We also study the possibility of PAC learning under $p$-tampering attacks in the \emph{non-targeted} (aka indiscriminate) setting where the adversary’s goal is to increase the risk of the generated hypothesis (for a random test example). We show that PAC learning is \emph{possible} under $p$-tampering poisoning attacks essentially whenever it is possible in the realizable setting without the attacks. We further show that PAC learning under ‘no-mistake’ adversarial noise is \emph{not} possible, if the adversary could choose the (still limited to only $p$ fraction of) tampered examples that she substitutes with adversarially chosen ones. Our formal model for such ‘bounded-budget’ tampering attackers is inspired by the notions of (strong) adaptive corruption in secure multi-party computation.

We study adversarial perturbations when the instances are uniformly distributed over {0,1}^n. We study both "inherent" bounds that apply to any problem and any classifier for such a problem as well as bounds that apply to specific problems and specific hypothesis classes. As the current literature contains multiple definitions of adversarial risk and robustness, we start by giving a taxonomy for these definitions based on their direct goals; we identify one of them as the one guaranteeing misclassification by pushing the instances to the error region. We then study some classic algorithms for learning monotone conjunctions and compare their adversarial risk and robustness under different definitions by attacking the hypotheses using instances drawn from the uniform distribution. We observe that sometimes these definitions lead to significantly different bounds. Thus, this study advocates for the use of the error-region definition, even though other definitions, in other contexts with context-dependent assumptions, may coincide with the error-region definition. Using the error-region definition of adversarial perturbations, we then study inherent bounds on risk and robustness of any classifier for any classification problem whose instances are uniformly distributed over {0,1}^n. Using the isoperimetric inequality for the Boolean hypercube, we show that for initial error 0.01, there always exists an adversarial perturbation that changes O(√n) bits of the instances to increase the risk to 0.5, making classifier's decisions meaningless. Furthermore, by also using the central limit theorem we show that when n→∞, at most c√n bits of perturbations, for a universal constant c<1.17, suffice for increasing the risk to 0.5, and the same c√n bits of perturbations on average suffice to increase the risk to 1, hence bounding the robustness by c√n.

The seminal result of Impagliazzo and Rudich (STOC 1989) gave a black-box separation between one-way functions and public-key encryption: a public-key encryption scheme cannot be constructed using one-way functions in a black-box way. In addition, their result implied black-box separations between one-way functions and protocols for certain Secure Function Evaluation (SFE) functionalities (in particular, Oblivious Transfer). Surprisingly, however, since then there has been no further progress in separating one-way functions and SFE functionalities. In this work, we present the complete picture for finite deterministic 2-party SFE functionalities, vis a vis one-way functions. We show that in case of semi-honest adversaries, one-way functions are black-box separated from all such SFE functionalities, except the ones which have unconditionally secure protocols (and hence do not rely on any computational hardness). In the case of active adversaries, a black-box one-way function is indeed useful for SFE, but we show that it is useful only as much as access to an ideal commitment functionality is useful.

Nonuniform proofs of security are common in cryptography, but traditional black-box separations consider only uniform security reductions. In this paper, we initiate a formal study of the power and limits of nonuniform black-box proofs of security. We first show that a known protocol (based on the existence of one-way permutations) that uses a nonuniform proof of security, and it cannot be proven secure through a uniform security reduction. Therefore, nonuniform proofs of security are indeed provably more powerful than uniform ones. We complement this result by showing that many known black-box separations in the uniform regime actually do extend to the nonuniform regime. We prove our results by providing general techniques for extending certain types of black-box separations to handle nonuniformity.

We construct a publicly verifiable protocol for proving computational work based on collision-resistant hash functions and a new plausible complexity assumption regarding the existence of "inherently sequential" hash functions. Our protocol is based on a novel construction of time-lock puzzles. Given a sampled "puzzle" P getsr Dn, where $n$ is the security parameter and Dn is the distribution of the puzzles, a corresponding "solution" can be generated using N evaluations of the sequential hash function, where N>n is another parameter, while any feasible adversarial strategy for generating valid solutions must take at least as much time as Ω(N) serial evaluations of the hash function after receiving $P$. Thus, valid solutions constitute a "proof" that Ω(N) parallel time elapsed since p was received. Solutions can be publicly and efficiently verified in time poly(n) ⋅ polylog(N). Applications of these "time-lock puzzles" include noninteractive timestamping of documents (where the distribution over the possible documents corresponds to the puzzle distribution Dn) and universally verifiable CPU benchmarks. Our construction is secure in the standard model under complexity assumptions (collision-resistant hash functions and inherently sequential hash functions), and makes black-box use of the underlying primitives. Consequently, the corresponding construction in the random oracle model is secure unconditionally. Moreover, as it is a public-coin protocol, it can be made non-interactive in the random oracle model using the Fiat-Shamir Heuristic.

We investigate the question of what languages can be decided efficiently with the help of a recursive collision-finding oracle. Such an oracle can be used to break collision-resistant hash functions or, more generally, statistically hiding commitments. The oracle we consider, Samdwhere d is the recursion depth, is based on the identically-named oracle defined in the work of Haitner et al. (FOCS '07). Our main result is a constant-round public-coin protocol "AM-Sam" that allows an efficient verifier to emulate a Samdoracle for any constant depth d = O(1) with the help of a BPPNPprover-AM-Sam allows us to conclude that if L is decidable by a k-adaptive randomized oracle algorithm with access to a SamO(1)oracle, then L ∈ AM[k] ∩ coAM[k]. The above yields the following corollary: assume there exists an O(1)-adaptive reduction that bases constant-round statistically hiding commitment on NP-hardness, then NP ⊆ coAM and the polynomial hierarchy collapses. The same result holds for any primitive that can be broken by SamO(1)including collision-resistant hash functions and O(1)-round oblivious transfer where security holds statistically for one of the parties. We also obtain non-trivial (though weaker) consequences for k-adaptive reductions for any k = poly(n). Prior to our work, most results in this research direction either applied only to non-adaptive reductions (Bogdanov and Trevisan, SIAM J. of Comp. '06 and Akavia et al., FOCS '06) or to one-way permutations (Brassard FOCS '79). The main technical tool we use to prove the above is a new constant-round public-coin protocol (SampleWithSize), which we believe to be of interest in its own right, that guarantees the following: given an efficient function f on n bits, let D be the output distribution D = f(Un), then SampleWithSize allows an efficient verifier Arthur to use an all-powerful prover Merlin's help to sample a random y ← D along with a good multiplicative approximation of the probability py= Pry' ← D[y' = y]. The crucial feature of SampleWithSize is that it extends even to distributions of the form D = f(Us), where Us is the uniform distribution on an efficiently decidable subset S ⊆ {0,1}n(such D are called efficiently samplable with post-selection), as long as the verifier is also given a good approximation of the value |S|.

We prove new results regarding the complexity of various complexity classes under randomized oracle reductions. We first prove that BPPPSZK⊆ AM ∩ coAM, where PSZK is the class of promise problems having statistical zero knowledge proofs. This strengthens the previously known facts that PSZK is closed under NC1truth-table reductions (Sahai and Vadhan, J. ACM '03) and that PPSZK⊆ AM ∩ coAM (Vadhan, personal communication). Our proof relies on showing that a certain class of real-valued functions that we call ℝ-TUAM can be approximated using an AM protocol. Then we investigate the power of randomized oracle reductions with relation to the notion of instance checking (Blum and Kannan, J. ACM '95). We observe that a theorem of Beigel implies that if any problem in TFNP such as Nash equilibrium is NP-hard under randomized oracle reductions, then SAT is checkable. We also observe that Beigel's theorem can be extended to an average-case setting by relating checking to the notion of program testing (Blum et al., JCSS '93). From this, we derive that if one-way functions can be based on NP-hardness via a randomized oracle reduction, then SAT is checkable. By showing that NP has a non-uniform tester, we also show that worst-case to average-case randomized oracle reduction for any relation (or language) R E NP implies that R has a nonuniform instance checker. These results hold even for adaptive randomized oracle reductions.