Version History: a conference version of this paper appeared in the Proceedings of the 18th International Workshop on Randomization and Computation (RANDOM'14). Full version posted as ECCC TR14-076 and arXiv:1405.7028 [cs.CC].

We present an explicit pseudorandom generator for oblivious, read-once, width-3 branching programs, which can read their input bits in any order. The generator has seed length \(Õ(\log^3 n)\).The previously best known seed length for this model is \(n^{1/2+o(1)}\) due to Impagliazzo, Meka, and Zuckerman (FOCS ’12). Our work generalizes a recent result of Reingold, Steinke, and Vadhan (RANDOM ’13) for permutation branching programs. The main technical novelty underlying our generator is a new bound on the Fourier growth of width-3, oblivious, read-once branching programs. Specifically, we show that for any \(f : \{0, 1\}^n → \{0, 1\}\) computed by such a branching program, and \(k ∈ [n]\),

 \(\displaystyle\sum_{s⊆[n]:|s|=k} \big| \hat{f}[s] \big | ≤n^2 ·(O(\log n))^k\),

where \(\hat{f}[s] = \mathbb{E}_U [f[U] \cdot (-1)^{s \cdot U}]\) is the standard Fourier transform over \(\mathbb{Z}^n_2\). The base \(O(\log n)\) of the Fourier growth is tight up to a factor of \(\log \log n\).

Building on the work of Daniely et al. (STOC 2014, COLT 2016), we study the connection between computationally efficient PAC learning and refutation of constraint satisfaction problems. Specifically, we prove that for every concept class \(\mathcal{P }\) , PAC-learning \(\mathcal{P}\) is polynomially equivalent to “random-right-hand-side-refuting” (“RRHS-refuting”) a dual class \(\mathcal{P}^∗ \), where RRHS-refutation of a class \(Q\) refers to refuting systems of equations where the constraints are (worst-case) functions from the class \( Q\) but the right-hand-sides of the equations are uniform and independent random bits. The reduction from refutation to PAC learning can be viewed as an abstraction of (part of) the work of Daniely, Linial, and Shalev-Schwartz (STOC 2014). The converse, however, is new, and is based on a combination of techniques from pseudorandomness (Yao ‘82) with boosting (Schapire ‘90). In addition, we show that PAC-learning the class of \(DNF\) formulas is polynomially equivalent to PAC-learning its dual class \(DNF ^∗\) , and thus PAC-learning \(DNF\) is equivalent to RRHS-refutation of \(DNF\) , suggesting an avenue to obtain stronger lower bounds for PAC-learning \(DNF\) than the quasipolynomial lower bound that was obtained by Daniely and Shalev-Schwartz (COLT 2016) assuming the hardness of refuting \(k\)-SAT.

Version History: 

We initiate the study of computational entropy in the quantum setting. We investigate to what extent the classical notions of computational entropy generalize to the quantum setting, and whether quantum analogues of classical theorems hold. Our main results are as follows. (1) The classical Leakage Chain Rule for pseudoentropy can be extended to the case that the leakage information is quantum (while the source remains classical). Specifically, if the source has pseudoentropy at least \(k\), then it has pseudoentropy at least \(k−ℓ \) conditioned on an \(ℓ \)-qubit leakage. (2) As an application of the Leakage Chain Rule, we construct the first quantum leakage-resilient stream-cipher in the bounded-quantum-storage model, assuming the existence of a quantum-secure pseudorandom generator. (3) We show that the general form of the classical Dense Model Theorem (interpreted as the equivalence between two definitions of pseudo-relative-min-entropy) does not extend to quantum states. Along the way, we develop quantum analogues of some classical techniques (e.g. the Leakage Simulation Lemma, which is proven by a Non-uniform Min-Max Theorem or Boosting). On the other hand, we also identify some classical techniques (e.g. Gap Amplification) that do not work in the quantum setting. Moreover, we introduce a variety of notions that combine quantum information and quantum complexity, and this raises several directions for future work. 

For Boolean functions computed by read-once, depth-D circuits with unbounded fan-in over the de Morgan basis, we present an explicit pseudorandom generator with seed length \(\tilde{O}(\log^{D+1} n)\). The previous best seed length known for this model was \(\tilde{O}(\log^{D+4} n)\), obtained by Trevisan and Xue (CCC ‘13) for all of AC0 (not just read-once). Our work makes use of Fourier analytic techniques for pseudorandomness introduced by Reingold, Steinke, and Vadhan (RANDOM ‘13) to show that the generator of Gopalan et al. (FOCS ‘12) fools read-once AC0. To this end, we prove a new Fourier growth bound for read-once circuits, namely that for every \(F : \{0,1\}^n\rightarrow \{0,1\}\) computed by a read-once, depth-\(D\) circuit,

\(\left|\hat{F}[s]\right| \leq O\left(\log^{D-1} n\right)^k,\)

where \(\hat{F}\) denotes the Fourier transform of \(F\) over \(\mathbb{Z}_2^n\).

Version History: Full version posted as https://arxiv.org/pdf/1308.5158.pdf.

We give two new characterizations of (\(\mathbb{F}_2\)-linear) locally testable error-correcting codes in terms of Cayley graphs over \(\mathbb{F}^h_2\) :

1. A locally testable code is equivalent to a Cayley graph over \(\mathbb{F}^h_2\) whose set of generators is significantly larger than \(h\) and has no short linear dependencies, but yields a shortest-path metric that embeds into \(\ell_1\) with constant distortion. This extends and gives a converse to a result of Khot and Naor (2006), which showed that codes with large dual distance imply Cayley graphs that have no low-distortion embeddings into \(\ell_1\).

2. A locally testable code is equivalent to a Cayley graph over \(\mathbb{F}^h_2\) that has significantly more than \(h\) eigenvalues near 1, which have no short linear dependencies among them and which “explain” all of the large eigenvalues. This extends and gives a converse to a recent construction of Barak et al. (2012), which showed that locally testable codes imply Cayley graphs that are small-set expanders but have many large eigenvalues.

Version History: Merge of conference papers from SODA ‘08 (with the same title) and RANDOM ‘08 (entitled “Tight Bounds for Hashing Block Sources”).

Hashing is fundamental to many algorithms and data structures widely used in practice. For the theoretical analysis of hashing, there have been two main approaches. First, one can assume that the hash function is truly random, mapping each data item independently and uniformly to the range. This idealized model is unrealistic because a truly random hash function requires an exponential number of bits (in the length of a data item) to describe. Alternatively, one can provide rigorous bounds on performance when explicit families of hash functions are used, such as 2-universal or \(O\)(1)-wise independent families. For such families, performance guarantees are often noticeably weaker than for ideal hashing.

In practice, however, it is commonly observed that simple hash functions, including 2-universal hash functions, perform as predicted by the idealized analysis for truly random hash functions. In this paper, we try to explain this phenomenon. We demonstrate that the strong performance of universal hash functions in practice can arise naturally from a combination of the randomness of the hash function and the data. Specifically, following the large body of literature on random sources and randomness extraction, we model the data as coming from a “block source,” whereby each new data item has some “entropy” given the previous ones. As long as the Rényi entropy per data item is sufficiently large, it turns out that the performance when choosing a hash function from a 2-universal family is essentially the same as for a truly random hash function. We describe results for several sample applications, including linear probing, chained hashing, balanced allocations, and Bloom filters.

Towards developing our results, we prove tight bounds for hashing block sources, determining the entropy required per block for the distribution of hashed values to be close to uniformly distributed.

 

Version History: Special Issue on STOC ‘10.

We give a new construction of pseudorandom generators from any one-way function. The construction achieves better parameters and is simpler than that given in the seminal work of Håstad, Impagliazzo, Levin, and Luby [SICOMP ’99]. The key to our construction is a new notion of next-block pseudoentropy, which is inspired by the notion of “in-accessible entropy” recently introduced in [Haitner, Reingold, Vadhan, and Wee, STOC ’09]. An additional advan- tage over previous constructions is that our pseudorandom generators are parallelizable and invoke the one-way function in a non-adaptive manner. Using [Applebaum, Ishai, and Kushilevitz, SICOMP ’06], this implies the existence of pseudorandom generators in NC\(^0\) based on the existence of one-way functions in NC\(^1\).

Version History: Preliminary version posted as arXiv:1003.4029 (Dec. 2010).

We study resilient functions and exposure‐resilient functions in the low‐entropy regime. A resilient function (a.k.a. deterministic extractor for oblivious bit‐fixing sources) maps any distribution on n‐bit strings in which k bits are uniformly random and the rest are fixed into an output distribution that is close to uniform. With exposure‐resilient functions, all the input bits are random, but we ask that the output be close to uniform conditioned on any subset of n – k input bits. In this paper, we focus on the case that k is sublogarithmic in n.

We simplify and improve an explicit construction of resilient functions for k sublogarithmic in n due to Kamp and Zuckerman (SICOMP 2006), achieving error exponentially small in krather than polynomially small in k. Our main result is that when k is sublogarithmic in n, the short output length of this construction (\(O(\log k)\) output bits) is optimal for extractors computable by a large class of space‐bounded streaming algorithms.

Next, we show that a random function is a resilient function with high probability if and only if k is superlogarithmic in n, suggesting that our main result may apply more generally. In contrast, we show that a random function is a static (resp. adaptive) exposure‐resilient function with high probability even if k is as small as a constant (resp. log log n). No explicit exposure‐resilient functions achieving these parameters are known. 

 

Version History: Preliminary version posted as Cryptology ePrint Archive Report 2011/553, under title “Non-Interactive Time-Stamping and Proofs of Work in the Random Oracle Model”.

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” \(\mathcal{P} \overset{$}\gets \mathbf{D}_n\), where \(n\) is the security parameter and \(\mathbf{D}_n\) 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 \(\Omega(N)\) sequential evaluations of the hash function after receiving \(\mathcal{P}\). Thus, valid solutions constitute a “proof” that \(\Omega(N)\) parallel time elapsed since \(\mathcal{P}\) was received. Solutions can be publicly and efficiently verified in time \(\mathrm{poly}(n) \cdot \mathrm{polylog}(N)\). Applications of these “time-lock puzzles” include noninteractive timestamping of documents (when the distribution over the possible documents corresponds to the puzzle distribution \(\mathbf{D}_n\)) 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.

Our construction makes a novel use of “depth-robust” directed acyclic graphs—ones whose depth remains large even after removing a constant fraction of vertices—which were previously studied for the purpose of complexity lower bounds. The construction bypasses a recent negative result of Mahmoody, Moran, and Vadhan (CRYPTO ‘11) for time-lock puzzles in the random oracle model, which showed that it is impossible to have time-lock puzzles like ours in the random oracle model if the puzzle generator also computes a solution together with the puzzle.

We present a new, more constructive proof of von Neumann’s Min-Max Theorem for two-player zero-sum game — specifically, an algorithm that builds a near-optimal mixed strategy for the second player from several best-responses of the second player to mixed strategies of the first player. The algorithm extends previous work of Freund and Schapire (Games and Economic Behavior ’99) with the advantage that the algorithm runs in poly\((n)\) time even when a pure strategy for the first player is a distribution chosen from a set of distributions over \(\{0,1\}^n\). This extension enables a number of additional applications in cryptography and complexity theory, often yielding uniform security versions of results that were previously only proved for nonuniform security (due to use of the non-constructive Min-Max Theorem).

We describe several applications, including a more modular and improved uniform version of Impagliazzo’s Hardcore Theorem (FOCS ’95), showing impossibility of constructing succinct non-interactive arguments (SNARGs) via black-box reductions under uniform hardness assumptions (using techniques from Gentry and Wichs (STOC ’11) for the nonuniform setting), and efficiently simulating high entropy distributions within any sufficiently nice convex set (extending a result of Trevisan, Tulsiani and Vadhan (CCC ’09)).

Version History: Full version posted as ECCC TR13-086 and arXiv:1306.3004 [cs.CC].

We present an explicit pseudorandom generator for oblivious, read-once, permutation branching programs of constant width that can read their input bits in any order. The seed length is \(O(\log^2n)\), where \(n\) is the length of the branching program. The previous best seed length known for this model was \(n^{1/2+o(1)}\), which follows as a special case of a generator due to Impagliazzo, Meka, and Zuckerman (FOCS 2012) (which gives a seed length of \(s^{1/2+o(1)}\) for arbitrary branching programs of size \(s\)). Our techniques also give seed length \(n^{1/2+o(1)}\) for general oblivious, read-once branching programs of width \(2^{n^{o(1)}}\)) , which is incomparable to the results of Impagliazzo et al.

Our pseudorandom generator is similar to the one used by Gopalan et al. (FOCS 2012) for read-once CNFs, but the analysis is quite different; ours is based on Fourier analysis of branching programs. In particular, we show that an oblivious, read-once, regular branching program of width \(w\) has Fourier mass at most \((2w^2)^k\) at level \(k\), independent of the length of the program.

We initiate a study of randomness condensers for sources that are efficiently samplable but may depend on the seed of the condenser. That is, we seek functions \(\mathsf{Cond} : \{0,1\}^n \times \{0,1\}^d \to \{0,1\}^m\)such that if we choose a random seed \(S \gets \{0,1\}^d\), and a source \(X = \mathcal{A}(S)\) is generated by a randomized circuit \(\mathcal{A}\) of size \(t\) such that \(X\) has min- entropy at least \(k\) given \(S\), then \(\mathsf{Cond}(X ; S)\) should have min-entropy at least some \(k'\) given \(S\). The distinction from the standard notion of randomness condensers is that the source \(X\) may be correlated with the seed \(S\) (but is restricted to be efficiently samplable). Randomness extractors of this type (corresponding to the special case where \(k' = m\)) have been implicitly studied in the past (by Trevisan and Vadhan, FOCS ‘00).

We show that:

• Unlike extractors, we can have randomness condensers for samplable, seed-dependent sources whose computational complexity is smaller than the size \(t\) of the adversarial sampling algorithm \(\mathcal{A}\). Indeed, we show that sufficiently strong collision-resistant hash functions are seed-dependent condensers that produce outputs with min-entropy \(k' = m – \mathcal{O}(\log t)\), i.e. logarithmic entropy deficiency.

• Randomness condensers suffice for key derivation in many cryptographic applications: when an adversary has negligible success probability (or negligible “squared advantage” [3]) for a uniformly random key, we can use instead a key generated by a condenser whose output has logarithmic entropy deficiency.

• Randomness condensers for seed-dependent samplable sources that are robust to side information generated by the sampling algorithm imply soundness of the Fiat-Shamir Heuristic when applied to any constant-round, public-coin interactive proof system.

Version History: Full version posted as ECCC TR11-141.

We provide a characterization of pseudoentropy in terms of hardness of sampling: Let (\(X, B\)) be jointly distributed random variables such that \(B\) takes values in a polynomial-sized set. We show that \(B\) is computationally indistinguishable from a random variable of higher Shannon entropy given \(X\) if and only if there is no probabilistic polynomial-time \(S\) such that \((X, S(X))\) has small KL divergence from \((X, B)\). This can be viewed as an analogue of the Impagliazzo Hard- core Theorem (FOCS ‘95) for Shannon entropy (rather than min-entropy).

Using this characterization, we show that if \(f\) is a one-way function, then \((f(U_n), U_n)\) has “next-bit pseudoentropy” at least \(n + \log n\), establishing a conjecture of Haitner, Reingold, and Vadhan (STOC ‘10). Plugging this into the construction of Haitner et al., this yields a simpler construction of pseudorandom generators from one-way functions. In particular, the construction only performs hashing once, and only needs the hash functions that are randomness extractors (e.g. universal hash functions) rather than needing them to support “local list-decoding” (as in the Goldreich–Levin hardcore predicate, STOC ‘89).

With an additional idea, we also show how to improve the seed length of the pseudorandom generator to \(\tilde{O}(n^3)\), compared to \(\tilde{O}(n^4)\) in the construction of Haitner et al.

In this work we revisit the question of basing cryptography on imperfect randomness. Bosley and Dodis (TCC’07) showed that if a source of randomness \(\mathcal{R}\) is “good enough” to generate a secret key capable of encrypting \(k\) bits, then one can deterministically extract nearly \(k\) almost uniform bits from \(\mathcal{R}\), suggesting that traditional privacy notions (namely, indistinguishability of encryption) requires an “extractable” source of randomness. Other, even stronger impossibility results are known for achieving privacy under specific “non-extractable” sources of randomness, such as the \(\gamma\)-Santha-Vazirani (SV) source, where each next bit has fresh entropy, but is allowed to have a small bias \(\gamma < 1\) (possibly depending on prior bits).

We ask whether similar negative results also hold for a more recent notion of privacy called differential privacy (Dwork et al., TCC’06), concentrating, in particular, on achieving differential privacy with the Santha-Vazirani source. We show that the answer is no. Specifically, we give a differentially private mechanism for approximating arbitrary “low sensitivity” functions that works even with randomness coming from a \(\gamma\)-Santha-Vazirani source, for any \(\gamma < 1\). This provides a somewhat surprising “separation” between traditional privacy and differential privacy with respect to imperfect randomness.

Interestingly, the design of our mechanism is quite different from the traditional “additive-noise” mechanisms (e.g., Laplace mechanism) successfully utilized to achieve differential privacy with perfect randomness. Indeed, we show that any (non-trivial) “SV-robust” mechanism for our problem requires a demanding property called consistent sampling, which is strictly stronger than differential privacy, and cannot be satisfied by any additive-noise mechanism.