We show new lower bounds and impossibility results for general (possibly <i>non-black-box</i>) zero-knowledge proofs and arguments. Our main results are that, under reasonable complexity assumptions:
<ol>
<li> There does not exist a two-round zero-knowledge <i>proof</i> system with perfect completeness for an NP-complete language. The previous impossibility result for two-round zero ... more >>>

Let $\cal C$ be a class of distributions over $\B^n$. A deterministic randomness extractor for $\cal C$ is a function $E:\B^n \ar \B^m$ such that for any $X$ in $\cal C$ the distribution $E(X)$ is statistically close to the uniform distribution. A long line of research deals with explicit constructions ... more >>>

We consider the problem of random selection, where $p$ players follow a protocol to jointly select a random element of a universe of size $n$. However, some of the players may be adversarial and collude to force the output to lie in a small subset of the universe. We describe ... more >>>

We give an explicit construction of a constant-distortion embedding of an n-dimensional L_2 space into an n^{1+o(1)}-dimensional L_1 space.

We give new constructions of randomness extractors and lossless condensers that are optimal to within constant factors in both the seed length and the output length. For extractors, this matches the parameters of the current best known construction [LRVW03]; for lossless condensers, the previous best constructions achieved optimality to within ... more >>>

We initiate a general study of the randomness complexity of
property testing, aimed at reducing the randomness complexity of
testers without (significantly) increasing their query complexity.
One concrete motovation for this study is provided by the
observation that the product of the randomness and query complexity
of a tester determine ... more >>>

Motivated by a recent study of Zimand (22nd CCC, 2007),
we consider the average-case complexity of property testing
(focusing, for clarity, on testing properties of Boolean strings).
We make two observations:

1) In the context of average-case analysis with respect to
the uniform distribution (on all strings of ... more >>>

We clarify the role of Kolmogorov complexity in the area of randomness extraction. We show that a computable function is an almost randomness extractor if and only if it is a Kolmogorov complexity
extractor, thus establishing a fundamental equivalence between two forms of extraction studied in the literature: Kolmogorov extraction
more >>>

Non-malleable coding, introduced by Dziembowski, Pietrzak and Wichs (ICS 2010), aims for protecting the integrity of information against tampering attacks in situations where error-detection is impossible. Intuitively, information encoded by a non-malleable code either decodes to the original message or, in presence of any tampering, to an unrelated message. Non-malleable ... more >>>

We consider randomness extraction by AC0 circuits. The main parameter, $n$, is the length of the source, and all other parameters are functions of it. The additional extraction parameters are the min-entropy bound $k=k(n)$, the seed length $r=r(n)$, the output length $m=m(n)$, and the (output) deviation bound $\epsilon=\epsilon(n)$.

For $k ... more >>>

The quantified derandomization problem of a circuit class $\mathcal{C}$ with a function $B:\mathbb{N}\rightarrow\mathbb{N}$ is the following: Given an input circuit $C\in\mathcal{C}$ over $n$ bits, deterministically distinguish between the case that $C$ accepts all but $B(n)$ of its inputs and the case that $C$ rejects all but $B(n)$ of its inputs. ... more >>>

The Erdos-Rado sunflower theorem (Journal of Lond. Math. Soc. 1960) is a fundamental result in combinatorics, and the corresponding sunflower conjecture is a central open problem. Motivated by applications in complexity theory, Rossman (FOCS 2010) extended the result to quasi-sunflowers, where similar conjectures emerge about the optimal parameters for which ... more >>>

We show that a small subset of seeds of any strong extractor also gives a strong extractor with similar parameters when the number of output bits is a constant. Specifically, if $Ext: \{0,1\}^n \times \{0,1\}^t \to \{0,1\}^m$ is a strong $(k,\epsilon)$-extractor, then for at least 99% of choices of $\tilde{O}(n ... more >>>

We revisit the fundamental problem of determining seed length lower bounds for strong extractors and natural variants thereof. These variants stem from a ``change in quantifiers'' over the seeds of the extractor: While a strong extractor requires that the average output bias (over all seeds) is small for all input ... more >>>

In this work we study bounded collusion protocols (BCPs) recently introduced in the context of secret sharing by Kumar, Meka, and Sahai (FOCS 2019). These are multi-party communication protocols on $n$ parties where in each round a subset of $p$-parties (the collusion bound) collude together and write a function of ... more >>>

Secret-sharing is one of the most basic and oldest primitives in cryptography, introduced by Shamir and Blakely in the 70s. It allows to strike a meaningful balance between availability and confidentiality of secret information. It has a host of applications most notably in threshold cryptography and multi-party computation. All known ... more >>>

At ITCS 2010, Dziembowski, Pietrzak, and Wichs introduced Non-malleable Codes (NMCs). Non-malleability is one of the strongest and most challenging notions of security considered in cryptography and protects against tampering attacks. In the context of coding schemes, non-malleability requires that it be infeasible to tamper the codeword of a message ... more >>>

We continue a line of work on extracting random bits from weak sources that are generated by simple processes. We focus on the model of locally samplable sources, where each bit in the source depends on a small number of (hidden) uniformly random input bits. Also known as local sources, ... more >>>

Randomness extractors provide a generic way of converting sources of randomness that are
merely unpredictable into almost uniformly random bits. While in general, deterministic randomness
extraction is impossible, it is possible if the source has some structural constraints.
While much of the literature on deterministic extraction has focused on sources ... more >>>

We study extractors computable in uniform $\mathrm{AC}^0$ and uniform $\mathrm{NC}^1$.

For the $\mathrm{AC}^0$ setting, we give a construction such that for every $k \ge n/ \mathrm{poly} \log n, \eps \ge 2^{-\mathrm{poly} \log n}$, it can extract $(1-\gamma)k$ randomness from an $(n, k)$ source for an arbitrary constant ... more >>>

We present the first efficient averaging sampler that achieves asymptotically optimal randomness complexity and near-optimal sample complexity for natural parameter choices. Specifically, for any constant $\alpha > 0$, for $\delta > 2^{-\mathrm{poly}(1 / \varepsilon)}$, it uses $m + O(\log (1 / \delta))$ random bits to output $t = O(\log(1 ... more >>>

Trevisan and Vadhan (FOCS 2000) introduced the notion of (seedless) extractors for samplable distributions as a possible solution to the problem of extracting random keys for cryptographic protocols from weak sources of randomness.
They showed that under a very strong complexity theoretic assumption, there exists a constant $\alpha>0$ such that ... more >>>

Seeded extractors are fundamental objects in pseudorandomness and cryptography, and a deep line of work has designed polynomial-time seeded extractors with nearly-optimal parameters. However, existing constructions of seeded extractors with short seed length and large output length run in time $\Omega(n \log(1/\varepsilon))$ and often slower, where $n$ is the input ... more >>>

Trevisan and Vadhan (FOCS 2000) introduced the notion of (seedless) extractors for samplable distributions. They showed that under a very strong complexity theoretic hardness assumption, there are extractors for samplable distributions with large min-entropy of $k=(1-\gamma) \cdot n$, for some small constant $\gamma>0$. Recent work by Ball, Goldin, Dachman-Soled and ... more >>>