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Electronic Colloquium on Computational Complexity

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All reports by Author Ronen Shaltiel:

TR20-094 | 24th June 2020
Ronen Shaltiel

Is it possible to improve Yao’s XOR lemma using reductions that exploit the efficiency of their oracle?

Yao's XOR lemma states that for every function $f:\set{0,1}^k \ar \set{0,1}$, if $f$ has hardness $2/3$ for $P/poly$ (meaning that for every circuit $C$ in $P/poly$, $\Pr[C(X)=f(X)] \le 2/3$ on a uniform input $X$), then the task of computing $f(X_1) \oplus \ldots \oplus f(X_t)$ for sufficiently large $t$ has hardness ... more >>>

TR20-047 | 16th April 2020
Ronen Shaltiel, Jad Silbak

Explicit Uniquely Decodable Codes for Space Bounded Channels That Achieve List-Decoding Capacity

We consider codes for space bounded channels. This is a model for communication under noise that was introduced by Guruswami and Smith (J. ACM 2016) and lies between the Shannon (random) and Hamming (adversarial) models. In this model, a channel is a space bounded procedure that reads the codeword in ... more >>>

TR19-090 | 27th June 2019
Ronen Shaltiel, Swastik Kopparty, Jad Silbak

Quasilinear time list-decodable codes for space bounded channels

Revisions: 2

We consider codes for space bounded channels. This is a model for communication under noise that was studied by Guruswami and Smith (J. ACM 2016) and lies between the Shannon (random) and Hamming (adversarial) models. In this model, a channel is a space bounded procedure that reads the codeword in ... more >>>

TR19-081 | 31st May 2019
Iftach Haitner, Noam Mazor, Ronen Shaltiel, Jad Silbak

Channels of Small Log-Ratio Leakage and Characterization of Two-Party Differentially Private Computation

Revisions: 1

Consider a PPT two-party protocol ?=(A,B) in which the parties get no private inputs and obtain outputs O^A,O^B?{0,1}, and let V^A and V^B denote the parties’ individual views. Protocol ? has ?-agreement if Pr[O^A=O^B]=1/2+?. The leakage of ? is the amount of information a party obtains about the event {O^A=O^B}; ... more >>>

TR18-071 | 15th April 2018
Iftach Haitner, Kobbi Nissim, Eran Omri, Ronen Shaltiel, Jad Silbak

Computational Two-Party Correlation

Revisions: 1

Let $\pi$ be an efficient two-party protocol that given security parameter $\kappa$, both parties output single bits $X_\kappa$ and $Y_\kappa$, respectively. We are interested in how $(X_\kappa,Y_\kappa)$ ``appears'' to an efficient adversary that only views the transcript $T_\kappa$. We make the following contributions:

\item We develop new tools to ... more >>>

TR18-061 | 6th April 2018
Aryeh Grinberg, Ronen Shaltiel, Emanuele Viola

Indistinguishability by adaptive procedures with advice, and lower bounds on hardness amplification proofs

Revisions: 5

We study how well can $q$-query decision trees distinguish between the
following two distributions: (i) $R=(R_1,\ldots,R_N)$ that are i.i.d.
variables, (ii) $X=(R|R \in A)$ where $A$ is an event s.t. $\Pr[R \in A] \ge
2^{-a}$. We prove two lemmas:

- Forbidden-set lemma: There exists $B \subseteq [N]$ of
size ... more >>>

TR16-134 | 29th August 2016
Ronen Shaltiel, Jad Silbak

Explicit List-Decodable Codes with Optimal Rate for Computationally Bounded Channels

Revisions: 1

A stochastic code is a pair of encoding and decoding procedures $(Enc,Dec)$ where $Enc:\{0,1\}^k \times \{0,1\}^d \to \{0,1\}^n$, and a message $m \in \{0,1\}^k$ is encoded by $Enc(m,S)$ where $S \from \{0,1\}^d$ is chosen uniformly by the encoder. The code is $(p,L)$-list-decodable against a class $\mathcal{C}$ of ``channel functions'' $C:\{0,1\}^n ... more >>>

TR16-037 | 15th March 2016
Sergei Artemenko, Russell Impagliazzo, Valentine Kabanets, Ronen Shaltiel

Pseudorandomness when the odds are against you

Impagliazzo and Wigderson showed that if $\text{E}=\text{DTIME}(2^{O(n)})$ requires size $2^{\Omega(n)}$ circuits, then
every time $T$ constant-error randomized algorithm can be simulated deterministically in time $\poly(T)$. However, such polynomial slowdown is a deal breaker when $T=2^{\alpha \cdot n}$, for a constant $\alpha>0$, as is the case for some randomized algorithms for ... more >>>

TR15-051 | 5th April 2015
Benny Applebaum, Sergei Artemenko, Ronen Shaltiel, Guang Yang

Incompressible Functions, Relative-Error Extractors, and the Power of Nondeterminsitic Reductions

Revisions: 2

A circuit $C$ \emph{compresses} a function $f:\{0,1\}^n\rightarrow \{0,1\}^m$ if given an input $x\in \{0,1\}^n$ the circuit $C$ can shrink $x$ to a shorter $\ell$-bit string $x'$ such that later, a computationally-unbounded solver $D$ will be able to compute $f(x)$ based on $x'$. In this paper we study the existence of ... more >>>

TR13-057 | 5th April 2013
Ruiwen Chen, Valentine Kabanets, Antonina Kolokolova, Ronen Shaltiel, David Zuckerman

Mining Circuit Lower Bound Proofs for Meta-Algorithms

We show that circuit lower bound proofs based on the method of random restrictions yield non-trivial compression algorithms for ``easy'' Boolean functions from the corresponding circuit classes. The compression problem is defined as follows: given the truth table of an $n$-variate Boolean function $f$ computable by some unknown small circuit ... more >>>

TR11-127 | 18th September 2011
Ronen Shaltiel

Dispersers for affine sources with sub-polynomial entropy

We construct an explicit disperser for affine sources over $\F_2^n$ with entropy $k=2^{\log^{0.9} n}=n^{o(1)}$. This is a polynomial time computable function $D:\F_2^n \ar \B$ such that for every affine space $V$ of $\F_2^n$ that has dimension at least $k$, $D(V)=\set{0,1}$. This improves the best previous construction of Ben-Sasson and Kopparty ... more >>>

TR11-016 | 7th February 2011
Sergei Artemenko, Ronen Shaltiel

Lower bounds on the query complexity of non-uniform and adaptive reductions showing hardness amplification

Revisions: 1

Hardness amplification results show that for every function $f$ there exists a function $Amp(f)$ such that the following holds: if every circuit of size $s$ computes $f$ correctly on at most a $1-\delta$ fraction of inputs, then every circuit of size $s'$ computes $Amp(f)$ correctly on at most a $1/2+\eps$ ... more >>>

TR10-186 | 2nd December 2010
Bill Fefferman, Ronen Shaltiel, Chris Umans, Emanuele Viola

On beating the hybrid argument

The {\em hybrid argument}
allows one to relate
the {\em distinguishability} of a distribution (from
uniform) to the {\em
predictability} of individual bits given a prefix. The
argument incurs a loss of a factor $k$ equal to the
bit-length of the
distributions: $\epsilon$-distinguishability implies only
$\epsilon/k$-predictability. ... more >>>

TR10-129 | 16th August 2010
Jeff Kinne, Dieter van Melkebeek, Ronen Shaltiel

Pseudorandom Generators, Typically-Correct Derandomization, and Circuit Lower Bounds

The area of derandomization attempts to provide efficient deterministic simulations of randomized algorithms in various algorithmic settings. Goldreich and Wigderson introduced a notion of "typically-correct" deterministic simulations, which are allowed to err on few inputs. In this paper we further the study of typically-correct derandomization in two ways.

First, we ... more >>>

TR10-037 | 8th March 2010
Boaz Barak, Guy Kindler, Ronen Shaltiel, Benny Sudakov, Avi Wigderson

Simulating Independence: New Constructions of Condensers, Ramsey Graphs, Dispersers, and Extractors

We present new explicit constructions of *deterministic* randomness extractors, dispersers and related objects. We say that a
distribution $X$ on binary strings of length $n$ is a
$\delta$-source if $X$ assigns probability at most $2^{-\delta n}$
to any string of length $n$. For every $\delta>0$ we construct the
following poly($n$)-time ... more >>>

TR07-130 | 3rd December 2007
Ronen Shaltiel, Emanuele Viola

Hardness amplification proofs require majority

Hardness amplification is the fundamental task of
converting a $\delta$-hard function $f : {0,1}^n ->
{0,1}$ into a $(1/2-\eps)$-hard function $Amp(f)$,
where $f$ is $\gamma$-hard if small circuits fail to
compute $f$ on at least a $\gamma$ fraction of the
inputs. Typically, $\eps,\delta$ are small (and
$\delta=2^{-k}$ captures the case ... more >>>

TR07-069 | 29th July 2007
Ronen Shaltiel, Chris Umans

Low-end uniform hardness vs. randomness tradeoffs for AM

In 1998, Impagliazzo and Wigderson proved a hardness vs. randomness tradeoff for BPP in the {\em uniform setting}, which was subsequently extended to give optimal tradeoffs for the full range of possible hardness assumptions by Trevisan and Vadhan (in a slightly weaker setting). In 2003, Gutfreund, Shaltiel and Ta-Shma proved ... more >>>

TR05-145 | 5th December 2005
Ronen Shaltiel

How to get more mileage from randomness extractors

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 >>>

TR05-109 | 28th September 2005
Ariel Gabizon, Ran Raz, Ronen Shaltiel

Deterministic Extractors for Bit-fixing Sources by Obtaining an Independent Seed

An $(n,k)$-bit-fixing source is a distribution $X$ over $\B^n$ such that
there is a subset of $k$ variables in $X_1,\ldots,X_n$ which are uniformly
distributed and independent of each other, and the remaining $n-k$ variables
are fixed. A deterministic bit-fixing source extractor is a function $E:\B^n
\ar \B^m$ which on ... more >>>

TR04-115 | 1st December 2004
Iftach Haitner, Ronen Shaltiel

Statistical Zero-Knowledge Arguments for NP Using Approximable-Preimage-Size One-Way Functions

A statistical zero knowledge argument for NP is a cryptographic primitive that allows a polynomial-time prover to convince another
polynomial-time verifier of the validity of an NP statement. It is guaranteed that even an infinitely powerful verifier does not learn any
additional information but the validity of the claim.

Naor ... more >>>

TR04-086 | 12th October 2004
Ronen Shaltiel, Chris Umans

Pseudorandomness for Approximate Counting and Sampling

We study computational procedures that use both randomness and nondeterminism. Examples are Arthur-Merlin games and approximate counting and sampling of NP-witnesses. The goal of this paper is to derandomize such procedures under the weakest possible assumptions.

Our main technical contribution allows one to ``boost'' a given hardness assumption. One special ... more >>>

TR01-009 | 5th January 2001
Ronen Shaltiel

Towards proving strong direct product theorems

A fundamental question of complexity theory is the direct product
question. Namely weather the assumption that a function $f$ is hard on
average for some computational class (meaning that every algorithm from
the class has small advantage over random guessing when computing $f$)
entails that computing $f$ on ... more >>>

TR00-059 | 11th August 2000
Omer Reingold, Ronen Shaltiel, Avi Wigderson

Extracting Randomness via Repeated Condensing

On an input probability distribution with some (min-)entropy
an {\em extractor} outputs a distribution with a (near) maximum
entropy rate (namely the uniform distribution).
A natural weakening of this concept is a condenser, whose
output distribution has a higher entropy rate than the
input distribution (without losing
much of ... more >>>

TR00-009 | 21st February 2000
Russell Impagliazzo, Ronen Shaltiel, Avi Wigderson

Extractors and pseudo-random generators with optimal seed length

We give the first construction of a pseudo-random generator with
optimal seed length that uses (essentially) arbitrary hardness.
It builds on the novel recursive use of the NW-generator in
a previous paper by the same authors, which produced many optimal
generators one of which was pseudo-random. This is achieved ... more >>>

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