# Computational Complexity

Steinke, Thomas, Salil Vadhan, and Andrew Wan. “Pseudorandomness and Fourier growth bounds for width 3 branching programs.” Theory of Computing – Special Issue on APPROX-RANDOM 2014 13, no. 12 (2017): 1-50. Publisher's VersionAbstract

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 $$\log \log n$$.

Vadhan., Salil P.On learning vs. refutation.” 30th Conference on Learning Theory (COLT 17), 2017, 65, 1835-1848. Publisher's VersionAbstract
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.
Murtagh, Jack, Omer Reingold, Aaron Sidford, and Salil Vadhan. “Derandomization beyond connectivity: Undirected Laplacian systems in nearly logarithmic space.58th Annual IEEE Symposium on Foundations of Computer Science (FOCS 17), 2017. Publisher's VersionAbstract
Version History
ArXiv, 15 August 2017 https://arxiv.org/abs/1708.04634

We give a deterministic $$\overline{O} (\log n)$$-space algorithm for approximately solving linear systems given by Laplacians of undirected graphs, and consequently also approximating hitting times, commute times, and escape probabilities for undirected graphs. Previously, such systems were known to be solvable by randomized algorithms using $$O(\log n)$$ space (Doron, Le Gall, and Ta-Shma, 2017) and hence by deterministic algorithms using $$O(\log^{3/2} n)$$  space (Saks and Zhou, FOCS 1995 and JCSS 1999).

Our algorithm combines ideas from time-efficient Laplacian solvers (Spielman and Teng, STOC '04; Peng and Spielman, STOC '14) with ideas used to show that Undirected S-T Connectivity is in deterministic logspace (Reingold, STOC '05 and JACM '08; Rozenman and Vadhan, RANDOM '05).
Chen, Yi-Hsiu, Kai-Min Chung, Ching-Yi Lai, Salil P. Vadhan, and Xiaodi Wu.Computational notions of quantum min-entropy.” In Poster presention at QIP 2017 and oral presentation at QCrypt 2017, 2017. Publisher's VersionAbstract

Version History

ArXiv v1, 24 April 2017 https://arxiv.org/abs/1704.07309v1
ArXiv v2, 25 April 2017 https://arxiv.org/abs/1704.07309v2
ArXiv v3, 9 September 2017 https://arxiv.org/abs/1704.07309v3
ArXiv v4, 5 October 2017 https://arxiv.org/abs/1704.07309v4

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.

Bun, Mark, Yi-Hsiu Chen, and Salil Vadhan. “Separating computational and statistical differential privacy in the client-server model.” In Martin Hirt and Adam D. Smith, editors, Proceedings of the 14th IACR Theory of Cryptography Conference (TCC `16-B). Lecture Notes in Computer Science. Springer Verlag, 31 October-3 November, 2016. Publisher's VersionAbstract

Version History: Full version posted on Cryptology ePrint Archive, Report 2016/820.

Differential privacy is a mathematical definition of privacy for statistical data analysis. It guarantees that any (possibly adversarial) data analyst is unable to learn too much information that is specific to an individual. Mironov et al. (CRYPTO 2009) proposed several computa- tional relaxations of differential privacy (CDP), which relax this guarantee to hold only against computationally bounded adversaries. Their work and subsequent work showed that CDP can yield substantial accuracy improvements in various multiparty privacy problems. However, these works left open whether such improvements are possible in the traditional client-server model of data analysis. In fact, Groce, Katz and Yerukhimovich (TCC 2011) showed that, in this setting, it is impossible to take advantage of CDP for many natural statistical tasks.

Our main result shows that, assuming the existence of sub-exponentially secure one-way functions and 2-message witness indistinguishable proofs (zaps) for NP, that there is in fact a computational task in the client-server model that can be efficiently performed with CDP, but is infeasible to perform with information-theoretic differential privacy.

Chen, Sitan, Thomas Steinke, and Salil P. Vadhan. “Pseudorandomness for read-once, constant-depth circuits.” CoRR, 2015, 1504.04675. Publisher's VersionAbstract

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 $$\mathbb{Z}_2^n$$.

Haitner, Iftach, Omer Reingold, and Salil Vadhan. “Efficiency improvements in constructing pseudorandom generators from one-way functions.” SIAM Journal on Computing 42, no. 3 (2013): 1405-1430. Publisher's VersionAbstract

Version HistorySpecial 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$$^1$$.

Mahmoody, Mohammad, Tal Moran, and Salil Vadhan. “Publicly verifiable proofs of sequential work.” In Innovations in Theoretical Computer Science (ITCS ‘13), 373-388. ACM, 2013. Publisher's VersionAbstract

Version HistoryPreliminary 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.

Rothblum, Guy N., Salil Vadhan, and Avi Wigderson. “Interactive proofs of proximity: delegating computation in sublinear time.” In Proceedings of the 45th Annual ACM Symposium on Theory of Computing (STOC ‘13), 793-802. New York, NY: ACM, 2013. Publisher's VersionAbstract

We study interactive proofs with sublinear-time verifiers. These proof systems can be used to ensure approximate correctness for the results of computations delegated to an untrusted server. Following the literature on property testing, we seek proof systems where with high probability the verifier accepts every input in the language, and rejects every input that is far from the language. The verifier’s query complexity (and computation complexity), as well as the communication, should all be sublinear. We call such a proof system an Interactive Proof of Proximity (IPP).

• On the positive side, our main result is that all languages in $$\mathcal{NC}$$ have Interactive Proofs of Proximity with roughly $$\sqrt{n}$$ query and communication and complexities, and $$\mathrm{polylog} (n)$$ communication rounds.

This is achieved by identifying a natural language, membership in an affine subspace (for a structured class of subspaces), that is complete for constructing interactive proofs of proximity, and providing efficient protocols for it. In building an IPP for this complete language, we show a tradeoff between the query and communication complexity and the number of rounds. For example, we give a 2-round protocol with roughly $$n^{3/4}$$ queries and communication.

• On the negative side, we show that there exist natural languages in $$\mathcal{NC}^1$$, for which the sum of queries and communication in any constant-round interactive proof of proximity must be polynomially related to n. In particular, for any 2-round protocol, the sum of queries and communication must be at least $$\tilde{\Omega}(\sqrt{n})$$.

• Finally, we construct much better IPPs for specific functions, such as bipartiteness on random or well-mixing graphs, and the majority function. The query complexities of these protocols are provably better (by exponential or polynomial factors) than what is possible in the standard property testing model, i.e. without a prover.

Vadhan, Salil, and Colin Jia Zheng. “A uniform min-max theorem with applications in cryptography.” In Ran Canetti and Juan Garay, editors, Advances in Cryptology—CRYPTO ‘13, Lecture Notes on Computer Science, 8042:93-110. Springer Verlag, Lecture Notes in Computer Science, 2013. Publisher's VersionAbstract
Version History:
Full version published on ECCC2013 and IACR ePrint 2013.

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)).

Reingold, Omer, Thomas Steinke, and Salil Vadhan. “Pseudorandomness for regular branching programs via Fourier analysis.” In Sofya Raskhodnikova and José Rolim, editors, Proceedings of the 17th International Workshop on Randomization and Computation (RANDOM ‘13), Lecture Notes in Computer Science, 8096:655-670. Springer-Verlag, 2013. Publisher's VersionAbstract

Version HistoryFull 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.

Dodis, Yevgeniy, Thomas Ristenpart, and Salil Vadhan. “Randomness condensers for efficiently samplable, seed-dependent sources.” In Ronald Cramer, editor, Proceedings of the 9th IACR Theory of Cryptography Conference (TCC ‘12), Lecture Notes on Computer Science, 7194:618-635. Springer-Verlag, 2012. Publisher's VersionAbstract

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.

Vadhan, Salil, and Colin Jia Zheng. “Characterizing pseudoentropy and simplifying pseudorandom generator constructions.” In Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC ‘12), 817-836. ACM, 2012. Publisher's VersionAbstract

Version HistoryFull 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.

Gopalan, Parikshit, Raghu Meka, Omer Reingold, Luca Tevisan, and Salil Vadhan. “Better pseudorandom generators via milder pseudorandom restrictions.” In Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS ‘12), 120-129. IEEE, 2012. Publisher's VersionAbstract

Version HistoryFull version posted as ECCC TR12-123 and as arXiv:1210.0049 [cs.CC].

We present an iterative approach to constructing pseudorandom generators, based on the repeated application of mild pseudorandom restrictions. We use this template to construct pseudorandom generators for combinatorial rectangles and read-once $$\mathsf{CNF}$$s and a hitting set generator for width-3 branching programs, all of which achieve near-optimal seed-length even in the low-error regime: We get seed-length $$\tilde{O}(\log(n/\epsilon))$$ for error $$\epsilon$$. Previously, only constructions with seed-length $$O(log^{3/2}n)$$ or $$O(log^2n)$$were known for these classes with error $$\epsilon = 1/\mathrm{poly}(n)$$. The (pseudo)random restrictions we use are milder than those typically used for proving circuit lower bounds in that we only set a constant fraction of the bits at a time. While such restrictions do not simplify the functions drastically, we show that they can be derandomized using small-bias spaces.

Kamp, Jesse, Anup Rao, Salil Vadhan, and David Zuckerman. “Deterministic extractors for small-space sources.” Journal of Computer and System Sciences 77, no. 1 (2011): 191-220. Publisher's VersionAbstract

Version History: Special issue to celebrate Richard Karp's Kyoto Prize. Extended abstract in STOC '06.

We give polynomial-time, deterministic randomness extractors for sources generated in small space, where we model space $$s$$ sources on$$\{0,1\}^n$$ as sources generated by width $$2^s$$ branching programs. Specifically, there is a constant $$η > 0$$ such that for any $$ζ > n^{−η}$$, our algorithm extracts $$m = (δ − ζ)n$$ bits that are exponentially close to uniform (in variation distance) from space $$s$$ sources with min-entropy $$δn$$, where $$s = Ω(ζ^ 3n)$$. Previously, nothing was known for $$δ \ll 1/2,$$, even for space $$0$$. Our results are obtained by a reduction to the class of total-entropy independent sources. This model generalizes both the well-studied models of independent sources and symbol-fixing sources. These sources consist of a set of $$r$$ independent smaller sources over $$\{0, 1\}^\ell$$, where the total min-entropy over all the smaller sources is $$k$$. We give deterministic extractors for such sources when $$k$$ is as small as $$\mathrm{polylog}(r)$$, for small enough $$\ell$$.