# Publications by Year: 2011

2011
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 $$\ell$$.

Chung, Kai-Min, Omer Reingold, and Salil Vadhan. “S-T connectivity on digraphs with a known stationary distribution.” In ACM Transactions on Algorithms. Vol. 7. 3rd ed. ACM, 2011. Publisher's VersionAbstract

Version history: Preliminary versions in CCC '07 and on ECCC (TR07-030).

We present a deterministic logspace algorithm for solving S-T Connectivity on directed graphs if: (i) we are given a stationary distribution of the random walk on the graph in which both of the input vertices $$s$$ and $$t$$ have nonnegligible probability mass and (ii) the random walk which starts at the source vertex $$s$$ has polynomial mixing time. This result generalizes the recent deterministic logspace algorithm for S-T Connectivity on undirected graphs [Reingold, 2008]. It identifies knowledge of the stationary distribution as the gap between the S-T Connectivity problems we know how to solve in logspace (L) and those that capture all of randomized logspace (RL).

Ullman, Jon, and Salil Vadhan. “PCPs and the hardness of generating synthetic data.” In Yuval Ishai, editor, Proceedings of the 8th IACR Theory of Cryptography Conference (TCC ‘11), Lecture Notes on Computer Science, 5978:572-587. Springer-Verlag, 2011. Publisher's VersionAbstract

Version HistoryFull version posted as ECCC TR10-017. Invited to J. Cryptology selected papers from TCC 2011.

Assuming the existence of one-way functions, we show that there is no polynomial-time, differentially private algorithm $$\mathcal{A}$$ that takes a database $$D ∈ ({0, 1}^d)^n$$ and outputs a “synthetic database” $$\hat{D}$$ all of whose two-way marginals are approximately equal to those of $$D$$. (A two-way marginal is the fraction of database rows $$x ∈ {0, 1}^d$$ with a given pair of values in a given pair of columns.) This answers a question of Barak et al. (PODS ‘07), who gave an algorithm running in time $$\mathrm{poly} (n, 2^d)$$.

Our proof combines a construction of hard-to-sanitize databases based on digital signatures (by Dwork et al., STOC ‘09) with encodings based on probabilistically checkable proofs.

We also present both negative and positive results for generating “relaxed” synthetic data, where the fraction of rows in $$D$$ satisfying a predicate $$c$$ are estimated by applying $$c$$ to each row of $$\hat{D}$$ and aggregating the results in some way.

Dvir, Zeev, Dan Gutfreund, Guy Rothblum, and Salil Vadhan. “On approximating the entropy of polynomial mappings.” In Proceedings of the Second Symposium on Innovations in Computer Science (ICS 2011), 460-475. Tsinghua University Press, 2011. Publisher's VersionAbstract

Version HistoryFull version posted as ECCC TR10-60.

We investigate the complexity of the following computational problem:

Polynomial Entropy Approximation (PEA): Given a low-degree polynomial mapping $$p : \mathbb{F}^n → \mathbb{F}^m$$, where F is a finite field, approximate the output entropy $$H(p(U_n))$$, where $$U_n$$ is the uniform distribution on $$\mathbb{F}^n$$ and $$H$$ may be any of several entropy measures.

We show:

• Approximating the Shannon entropy of degree 3 polynomials $$p : \mathbb{F}_2^n \to \mathbb{F}^m_2$$ over $$\mathbb{F}_2$$ to within an additive constant (or even $$n^.9$$) is complete for $$\mathbf{SZKP_L}$$, the class of problems having statistical zero-knowledge proofs where the honest verifier and its simulator are computable in logarithmic space. ($$\mathbf{SZKP_L}$$contains most of the natural problems known to be in the full class $$\mathbf{SZKP}$$.)

• For prime fields $$\mathbb{F} \neq \mathbb{F}_2$$ and homogeneous quadratic polynomials $$p : \mathbb{F}^n \to \mathbb{F}^m$$, there is a probabilistic polynomial-time algorithm that distinguishes the case that $$p(U_n)$$) has entropy smaller than k from the case that $$p(U_n))$$ has min-entropy (or even Renyi entropy) greater than $$(2 + o(1))k$$.

• For degree d polynomials $$p : \mathbb{F}^n_2 \to \mathbb{F}^m_2$$ , there is a polynomial-time algorithm that distinguishes the case that $$p(U_n)$$ has max-entropy smaller than $$k$$ (where the max-entropy of a random variable is the logarithm of its support size) from the case that $$p(U_n)$$ has max-entropy at least $$(1 + o(1)) \cdot k^d$$ (for fixed $$d$$ and large $$k$$).

Mahmoody, Mohammad, Tal Moran, and Salil Vadhan. “Time-lock puzzles in the random oracle model.” In P. Rogaway, editor, Advances in Cryptology—CRYPTO ‘11, Lecture Notes in Computer Science, 6841:39-50. Springer-Verlag, 2011. Publisher's VersionAbstract

A time-lock puzzle is a mechanism for sending messages “to the future”. The sender publishes a puzzle whose solution is the message to be sent, thus hiding it until enough time has elapsed for the puzzle to be solved. For time-lock puzzles to be useful, generating a puzzle should take less time than solving it. Since adversaries may have access to many more computers than honest solvers, massively parallel solvers should not be able to produce a solution much faster than serial ones.

To date, we know of only one mechanism that is believed to satisfy these properties: the one proposed by Rivest, Shamir and Wagner (1996), who originally introduced the notion of time-lock puzzles. Their puzzle is based on the serial nature of exponentiation and the hardness of factoring, and is therefore vulnerable to advances in factoring techniques (as well as to quantum attacks).

In this work, we study the possibility of constructing time-lock puzzles in the random-oracle model. Our main result is negative, ruling out time-lock puzzles that require more parallel time to solve than the total work required to generate a puzzle. In particular, this should rule out black-box constructions of such time-lock puzzles from one-way permutations and collision-resistant hash-functions. On the positive side, we construct a time-lock puzzle with a linear gap in parallel time: a new puzzle can be generated with one round of $${n}$$ parallel queries to the random oracle, but $${n}$$ rounds of serial queries are required to solve it (even for massively parallel adversaries).