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All reports by Author Eli Ben-Sasson:

TR17-134 | 8th September 2017
Eli Ben-Sasson, Iddo Bentov, Ynon Horesh, Michael Riabzev

Fast Reed-Solomon Interactive Oracle Proofs of Proximity

The family of Reed-Solomon (RS) codes plays a prominent role in the construction of quasilinear probabilistically checkable proofs (PCPs) and interactive oracle proofs (IOPs) with perfect zero knowledge and polylogarithmic verifiers. The large concrete computational complexity required to prove membership in RS codes is one of the biggest obstacles to ... more >>>

TR16-156 | 12th October 2016
Eli Ben-Sasson, Alessandro Chiesa, Michael Forbes, Ariel Gabizon, Michael Riabzev, Nicholas Spooner

On Probabilistic Checking in Perfect Zero Knowledge

Revisions: 1

We present the first constructions of *single*-prover proof systems that achieve *perfect* zero knowledge (PZK) for languages beyond NP, under no intractability assumptions:

1. The complexity class #P has PZK proofs in the model of Interactive PCPs (IPCPs) [KR08], where the verifier first receives from the prover a PCP and ... more >>>

TR16-149 | 23rd September 2016
Eli Ben-Sasson, iddo Ben-Tov, Ariel Gabizon, Michael Riabzev

A security analysis of Probabilistically Checkable Proofs

Comments: 1

Probabilistically Checkable Proofs (PCPs) [Babai et al. FOCS 90; Arora et al. JACM 98] can be used to construct asymptotically efficient cryptographic zero knowledge arguments of membership in any language in NEXP, with minimal communication complexity and computational effort on behalf of both prover and verifier [Babai et al. STOC ... more >>>

TR16-073 | 7th May 2016
Eli Ben-Sasson, iddo Ben-Tov, Ariel Gabizon, Michael Riabzev

Improved concrete efficiency and security analysis of Reed-Solomon PCPPs

Revisions: 1 , Comments: 1

A Probabilistically Checkable Proof of Proximity (PCPP) for a linear code $C$, enables to determine very efficiently if a long input $x$, given as an oracle, belongs to $C$ or is far from $C$.
PCPPs are often a central component of constructions of Probabilistically Checkable Proofs (PCP)s [Babai et al. ... more >>>

TR16-046 | 23rd March 2016
Eli Ben-Sasson, Alessandro Chiesa, Ariel Gabizon, Michael Riabzev, Nicholas Spooner

Short Interactive Oracle Proofs with Constant Query Complexity, via Composition and Sumcheck

Revisions: 2

We study *interactive oracle proofs* (IOPs) (Ben-Sasson, Chiesa, Spooner '16), which combine aspects of probabilistically checkable proofs (PCPs) and interactive proofs (IPs). We present IOP constructions and general techniques that enable us to obtain tradeoffs in proof length versus query complexity that are not known to be achievable via PCPs ... more >>>

TR16-001 | 9th January 2016
Eli Ben-Sasson, Alessandro Chiesa, Ariel Gabizon, Madars Virza

Quasi-Linear Size Zero Knowledge from Linear-Algebraic PCPs

Revisions: 1

The seminal result that every language having an interactive proof also has a zero-knowledge interactive proof assumes the existence of one-way functions. Ostrovsky and Wigderson (ISTCS 1993) proved that this assumption is necessary: if one-way functions do not exist, then only languages in BPP have zero-knowledge interactive proofs.

Ben-Or et ... more >>>

TR15-209 | 29th December 2015
Eli Ben-Sasson, Gal Maor

On the information leakage of public-output protocols

In this paper three complexity measures are studied: (i) internal information, (ii) external information, and (iii) a measure called here "output information". Internal information (i) measures the counter-party privacy-loss inherent in a communication protocol. Similarly, the output information (iii) measures the reduction in input-privacy that is inherent when the output ... more >>>

TR15-139 | 25th August 2015
Eli Ben-Sasson, Gal Maor

Lower bound for communication complexity with no public randomness

We give a self contained proof of a logarithmic lower bound on the communication complexity of any non redundant function, given that there is no access to shared randomness. This bound was first stated in Yao's seminal paper [STOC 1979], but no full proof appears in the literature.

Our proof ... more >>>

TR15-094 | 10th June 2015
Eli Ben-Sasson, iddo Ben-Tov, Ivan Bjerre Damgard, Yuval Ishai, Noga Ron-Zewi

On Public Key Encryption from Noisy Codewords

Several well-known public key encryption schemes, including those of Alekhnovich (FOCS 2003), Regev (STOC 2005), and Gentry, Peikert and Vaikuntanathan (STOC 2008), rely on the conjectured intractability of inverting noisy linear encodings. These schemes are limited in that they either require the underlying field to grow with the security parameter, ... more >>>

TR14-017 | 9th February 2014
Eli Ben-Sasson, Emanuele Viola

Short PCPs with projection queries

We construct a PCP for NTIME(2$^n$) with constant
soundness, $2^n \poly(n)$ proof length, and $\poly(n)$
queries where the verifier's computation is simple: the
queries are a projection of the input randomness, and the
computation on the prover's answers is a 3CNF. The
previous upper bound for these two computations was
more >>>

TR13-085 | 13th June 2013
Eli Ben-Sasson, Yohay Kaplan, Swastik Kopparty, Or Meir, Henning Stichtenoth

Constant rate PCPs for circuit-SAT with sublinear query complexity

The PCP theorem (Arora et. al., J. ACM 45(1,3)) says that every NP-proof can be encoded to another proof, namely, a probabilistically checkable proof (PCP), which can be tested by a verifier that queries only a small part of the PCP. A natural question is how large is the blow-up ... more >>>

TR12-159 | 20th November 2012
Eli Ben-Sasson, Michael Viderman

A Combinatorial Characterization of smooth LTCs and Applications

The study of locally testable codes (LTCs) has benefited from a number of nontrivial constructions discovered in recent years. Yet we still lack a good understanding of what makes a linear error correcting code locally testable and as a result we do not know what is the rate-limit of LTCs ... more >>>

TR12-148 | 7th November 2012
Eli Ben-Sasson, Ariel Gabizon, Yohay Kaplan, Swastik Kopparty, Shubhangi Saraf

A new family of locally correctable codes based on degree-lifted algebraic geometry codes

Revisions: 1

We describe new constructions of error correcting codes, obtained by "degree-lifting" a short algebraic geometry (AG) base-code of block-length $q$ to a lifted-code of block-length $q^m$, for arbitrary integer $m$. The construction generalizes the way degree-$d$, univariate polynomials evaluated over the $q$-element field (also known as Reed-Solomon codes) are "lifted" ... more >>>

TR12-135 | 26th October 2012
Eli Ben-Sasson, Noga Ron-Zewi, Madhur Tulsiani, Julia Wolf

Sampling-based proofs of almost-periodicity results and algorithmic applications

Revisions: 2

We give new combinatorial proofs of known almost-periodicity results for sumsets of sets with small doubling in the spirit of Croot and Sisask [Geom. Funct. Anal. 2010], whose almost-periodicity lemma has had far-reaching implications in additive combinatorics. We provide an alternative (and $L^p$-norm free) point of view, which allows for ... more >>>

TR12-049 | 27th April 2012
Eli Ben-Sasson, Noga Ron-Zewi, Madhu Sudan

Sparse affine-invariant linear codes are locally testable

We show that sparse affine-invariant linear properties over arbitrary finite fields are locally testable with a constant number of queries. Given a finite field ${\mathbb{F}}_q$ and an extension field ${\mathbb{F}}_{q^n}$, a property is a set of functions mapping ${\mathbb{F}}_{q^n}$ to ${\mathbb{F}}_q$. The property is said to be affine-invariant if it ... more >>>

TR12-045 | 22nd April 2012
Eli Ben-Sasson, Alessandro Chiesa, Daniel Genkin, Eran Tromer

On the Concrete-Efficiency Threshold of Probabilistically-Checkable Proofs

Revisions: 3

Probabilistically-Checkable Proofs (PCPs) form the algorithmic core that enables succinct verification of long proofs/computations in many cryptographic constructions, such as succinct arguments and proof-carrying data.

Despite the wonderful asymptotic savings they bring, PCPs are also the infamous computational bottleneck preventing these cryptographic constructions from being used in practice. This reflects ... more >>>

TR11-157 | 25th November 2011
Eli Ben-Sasson, Shachar Lovett, Noga Ron-Zewi

An additive combinatorics approach to the log-rank conjecture in communication complexity

Revisions: 1

For a {0,1}-valued matrix $M$ let CC($M$) denote the deterministic communication complexity of the boolean function associated with $M$. The log-rank conjecture of Lovasz and Saks [FOCS 1988] states that CC($M$) is at most $\log^c({\mbox{rank}}(M))$ for some absolute constant $c$ where rank($M$) denotes the rank of $M$ over the field ... more >>>

TR11-129 | 22nd September 2011
Eli Ben-Sasson, Ariel Gabizon

Extractors for Polynomials Sources over Constant-Size Fields of Small Characteristic

Let $F$ be the field of $q$ elements, where $q=p^{\ell}$ for prime $p$. Informally speaking, a polynomial source is a distribution over $F^n$ sampled by low degree multivariate polynomials. In this paper, we construct extractors for polynomial sources over fields of constant size $q$ assuming $p \ll q$.

More generally, ... more >>>

TR11-079 | 9th May 2011
Eli Ben-Sasson, Elena Grigorescu, Ghid Maatouk, Amir Shpilka, Madhu Sudan

On Sums of Locally Testable Affine Invariant Properties

Affine-invariant properties are an abstract class of properties that generalize some
central algebraic ones, such as linearity and low-degree-ness, that have been
studied extensively in the context of property testing. Affine invariant properties
consider functions mapping a big field $\mathbb{F}_{q^n}$ to the subfield $\mathbb{F}_q$ and include all
properties that form ... more >>>

TR11-070 | 1st May 2011
Eli Ben-Sasson, Michael Viderman

Composition of semi-LTCs by two-wise Tensor Products

In this paper we obtain a composition theorem that allows us to construct locally testable codes (LTCs) by repeated two-wise tensor products. This is the First composition theorem showing that repeating the two-wise tensor operation any constant number of times still results in a locally testable code, improving upon previous ... more >>>

TR10-200 | 14th December 2010
Eli Ben-Sasson, Michael Viderman

Towards lower bounds on locally testable codes via density arguments

The main open problem in the area of locally testable codes (LTCs) is whether there exists an asymptotically good family of LTCs and to resolve this question it suffices to consider the case of query complexity $3$. We argue that to refute the existence of such an asymptotically good family ... more >>>

TR10-199 | 14th December 2010
Eli Ben-Sasson, Ghid Maatouk, Amir Shpilka, Madhu Sudan

Symmetric LDPC codes are not necessarily locally testable

Locally testable codes, i.e., codes where membership in the code is testable with a constant number of queries, have played a central role in complexity theory. It is well known that a code must be a "low-density parity check" (LDPC) code for it to be locally testable, but few LDPC ... more >>>

TR10-144 | 20th September 2010
Eli Ben-Sasson, Noga Ron-Zewi

From Affine to Two-Source Extractors via Approximate Duality

Revisions: 1

Two-source and affine extractors and dispersers are fundamental objects studied in the context of derandomization. This paper shows how to construct two-source extractors and dispersers for arbitrarily small min-entropy rate in a black-box manner from affine extractors with sufficiently good parameters. Our analysis relies on the study of approximate duality, ... more >>>

TR10-125 | 11th August 2010
Eli Ben-Sasson, Jakob Nordström

Understanding Space in Proof Complexity: Separations and Trade-offs via Substitutions

For current state-of-the-art satisfiability algorithms based on the DPLL procedure and clause learning, the two main bottlenecks are the amounts of time and memory used. In the field of proof complexity, these resources correspond to the length and space of resolution proofs for formulas in conjunctive normal form (CNF). There ... more >>>

TR10-123 | 4th August 2010
Eli Ben-Sasson

Limitation on the rate of families of locally testable codes

Revisions: 1

This paper describes recent results which revolve around the question of the rate attainable by families of error correcting codes that are locally testable. Emphasis is placed on motivating the problem of proving upper bounds on the rate of these codes and a number of interesting open questions for future ... more >>>

TR10-108 | 9th July 2010
Eli Ben-Sasson, Madhu Sudan

Limits on the rate of locally testable affine-invariant codes

A linear code is said to be affine-invariant if the coordinates of the code can be viewed as a vector space and the code is invariant under an affine transformation of the coordinates. A code is said to be locally testable if proximity of a received word
to the code ... more >>>

TR10-085 | 20th May 2010
Eli Ben-Sasson, Jan Johannsen

Lower bounds for width-restricted clause learning on small width formulas

It has been observed empirically that clause learning does not significantly improve the performance of a SAT solver when restricted
to learning clauses of small width only. This experience is supported by lower bound theorems. It is shown that lower bounds on the runtime of width-restricted clause learning follow from ... more >>>

TR10-068 | 15th April 2010
Shir Ben-Israel, Eli Ben-Sasson, David Karger

Breaking local symmetries can dramatically reduce the length of propositional refutations

This paper shows that the use of ``local symmetry breaking'' can dramatically reduce the length of propositional refutations. For each of the three propositional proof systems known as (i) treelike resolution, (ii) resolution, and (iii) k-DNF resolution, we describe families of unsatisfiable formulas in conjunctive normal form (CNF) that are ... more >>>

TR10-044 | 12th March 2010
Eli Ben-Sasson, Swastik Kopparty

Affine Dispersers from Subspace Polynomials

{\em Dispersers} and {\em extractors} for affine sources of dimension $d$ in $\mathbb F_p^n$ --- where $\mathbb F_p$ denotes the finite field of prime size $p$ --- are functions $f: \mathbb F_p^n \rightarrow \mathbb F_p$ that behave pseudorandomly when their domain is restricted to any particular affine space $S \subseteq ... more >>>

TR10-004 | 6th January 2010
Eli Ben-Sasson, Michael Viderman

Low Rate Is Insufficient for Local Testability

Revisions: 3

Locally testable codes (LTCs) are error-correcting codes for which membership of a given word in the code can be tested probabilistically by examining it in very few locations.
Kaufman and Sudan \cite{KS07} proved that sparse, low-bias linear codes are locally testable (in particular sparse random codes are locally testable).
Kopparty ... more >>>

TR09-126 | 26th November 2009
Eli Ben-Sasson, Venkatesan Guruswami, Tali Kaufman, Madhu Sudan, Michael Viderman

Locally Testable Codes Require Redundant Testers

Revisions: 3

Locally testable codes (LTCs) are error-correcting codes for which membership, in the code, of a given word can be tested by examining it in very few locations. Most known constructions of locally testable codes are linear codes, and give error-correcting codes
whose duals have (superlinearly) {\em many} small weight ... more >>>

TR09-047 | 20th April 2009
Eli Ben-Sasson, Jakob Nordström

A Space Hierarchy for k-DNF Resolution

Comments: 1

The k-DNF resolution proof systems are a family of systems indexed by
the integer k, where the kth member is restricted to operating with
formulas in disjunctive normal form with all terms of bounded arity k
(k-DNF formulas). This family was introduced in [Krajicek 2001] as an
extension of the ... more >>>

TR09-034 | 25th March 2009
Eli Ben-Sasson, Jakob Nordström

Understanding Space in Resolution: Optimal Lower Bounds and Exponential Trade-offs

Comments: 1

For current state-of-the-art satisfiability algorithms based on the
DPLL procedure and clause learning, the two main bottlenecks are the
amounts of time and memory used. Understanding time and memory
consumption, and how they are related to one another, is therefore a
question of considerable practical importance. In the field of ... more >>>

TR09-007 | 9th January 2009
Eli Ben-Sasson, Michael Viderman

Tensor Products of Weakly Smooth Codes are Robust

We continue the study of {\em robust} tensor codes and expand the
class of base codes that can be used as a starting point for the
construction of locally testable codes via robust two-wise tensor
products. In particular, we show that all unique-neighbor expander
codes and all locally correctable codes, ... more >>>

TR09-002 | 23rd November 2008
Eli Ben-Sasson, Jakob Nordström

Short Proofs May Be Spacious: An Optimal Separation of Space and Length in Resolution

A number of works have looked at the relationship between length and space of resolution proofs. A notorious question has been whether the existence of a short proof implies the existence of a proof that can be verified using limited space.

In this paper we resolve the question by answering ... more >>>

TR07-127 | 22nd November 2007
Arie Matsliah, Eli Ben-Sasson, Prahladh Harsha, Oded Lachish

Sound 3-query PCPPs are Long

We initiate the study of the tradeoff between the {\em length} of a
probabilistically checkable proof of proximity (PCPP) and the
maximal {\em soundness} that can be guaranteed by a $3$-query
verifier with oracle access to the proof. Our main observation is
that a verifier limited to querying a short ... more >>>

TR04-060 | 22nd July 2004
Eli Ben-Sasson, Madhu Sudan

Simple PCPs with Poly-log Rate and Query Complexity

We give constructions of PCPs of length n*polylog(n) (with respect
to circuits of size n) that can be verified by making polylog(n)
queries to bits of the proof. These PCPs are not only shorter than
previous ones, but also simpler. Our (only) building blocks are
Reed-Solomon codes and the bivariate ... more >>>

TR04-046 | 4th June 2004
Eli Ben-Sasson, Madhu Sudan

Robust Locally Testable Codes and Products of Codes

We continue the investigation of locally testable codes, i.e.,
error-correcting codes for whom membership of a given word in the
code can be tested probabilistically by examining it in very few
locations. We give two general results on local testability:
First, motivated by the recently proposed notion of robust
probabilistically ... more >>>

TR04-021 | 23rd March 2004
Eli Ben-Sasson, Oded Goldreich, Prahladh Harsha, Madhu Sudan, Salil Vadhan

Robust PCPs of Proximity, Shorter PCPs and Applications to Coding

We continue the study of the trade-off between the length of PCPs
and their query complexity, establishing the following main results
(which refer to proofs of satisfiability of circuits of size $n$):
We present PCPs of length $\exp(\tildeO(\log\log n)^2)\cdot n$
that can be verified by making $o(\log\log n)$ Boolean queries.
more >>>

TR04-016 | 3rd March 2004
Michael Alekhnovich, Eli Ben-Sasson

Linear Upper Bounds for Random Walk on Small Density Random 3CNFs

We analyze the efficiency of the random walk algorithm on random 3CNF instances, and prove em linear upper bounds on the running time
of this algorithm for small clause density, less than 1.63. Our upper bound matches the observed running time to within a multiplicative factor. This is the ... more >>>

TR03-019 | 3rd April 2003
Eli Ben-Sasson, Oded Goldreich, Madhu Sudan

Bounds on 2-Query Codeword Testing.

Revisions: 1

We present upper bounds on the size of codes that are locally
testable by querying only two input symbols. For linear codes, we
show that any $2$-locally testable code with minimal distance
$\delta n$ over a finite field $F$ cannot have more than
$|F|^{3/\delta}$ codewords. This result holds even ... more >>>

TR03-006 | 23rd January 2003
Eli Ben-Sasson, Prahladh Harsha, Sofya Raskhodnikova

3CNF Properties are Hard to Test

For a boolean formula \phi on n variables, the associated property
P_\phi is the collection of n-bit strings that satisfy \phi. We prove
that there are 3CNF properties that require a linear number of queries,
even for adaptive tests. This contrasts with 2CNF properties
that are testable with O(\sqrt{n}) ... more >>>

TR03-004 | 24th December 2002
Eli Ben-Sasson, Prahladh Harsha

Lower Bounds for Bounded-Depth Frege Proofs via Buss-Pudlack Games

We present a simple proof of the bounded-depth Frege lower bounds of
Pitassi et. al. and Krajicek et. al. for the pigeonhole
principle. Our method uses the interpretation of proofs as two player
games given by Pudlak and Buss. Our lower bound is conceptually
simpler than previous ones, and relies ... more >>>

TR02-003 | 24th December 2001
Eli Ben-Sasson, Yonatan Bilu

A Gap in Average Proof Complexity

We present the first example of a natural distribution on instances
of an NP-complete problem, with the following properties.
With high probability a random formula from this
distribution (a) is unsatisfiable,
(b) has a short proof that can be found easily, and (c) does not have a short
(general) resolution ... more >>>

TR01-031 | 5th April 2001
Eli Ben-Sasson, Nicola Galesi

Space Complexity of Random Formulae in Resolution

We study the space complexity of refuting unsatisfiable random $k$-CNFs in
the Resolution proof system. We prove that for any large enough $\Delta$,
with high probability a random $k$-CNF over $n$ variables and $\Delta n$
clauses requires resolution clause space of
$\Omega(n \cdot \Delta^{-\frac{1+\epsilon}{k-2-\epsilon}})$,
for any $0<\epsilon<1/2$. For constant $\Delta$, ... more >>>

TR00-023 | 11th May 2000
Michael Alekhnovich, Eli Ben-Sasson, Alexander Razborov, Avi Wigderson

Pseudorandom Generators in Propositional Proof Complexity

We call a pseudorandom generator $G_n:\{0,1\}^n\to \{0,1\}^m$ {\em
hard} for a propositional proof system $P$ if $P$ can not efficiently
prove the (properly encoded) statement $G_n(x_1,\ldots,x_n)\neq b$ for
{\em any} string $b\in\{0,1\}^m$. We consider a variety of
``combinatorial'' pseudorandom generators inspired by the
Nisan-Wigderson generator on the one hand, and ... more >>>

TR00-005 | 17th January 2000
Eli Ben-Sasson, Russell Impagliazzo, Avi Wigderson

Near-Optimal Separation of Treelike and General Resolution

We present the best known separation
between tree-like and general resolution, improving
on the recent $\exp(n^\epsilon)$ separation of \cite{BEGJ98}.
This is done by constructing a natural family of contradictions, of
size $n$, that have $O(n)$-size resolution
refutations, but only $\exp (\Omega(n/\log n))$-size tree-like refutations.
This result ... more >>>

TR99-040 | 20th October 1999
Michael Alekhnovich, Eli Ben-Sasson, Alexander Razborov, Avi Wigderson

Space Complexity in Propositional Calculus

We study space complexity in the framework of
propositional proofs. We consider a natural model analogous to
Turing machines with a read-only input tape, and such
popular propositional proof systems as Resolution, Polynomial
Calculus and Frege systems. We propose two different space measures,
corresponding to the maximal number of bits, ... more >>>

TR99-022 | 14th June 1999
Eli Ben-Sasson, Avi Wigderson

Short Proofs are Narrow - Resolution made Simple

The width of a Resolution proof is defined to be the maximal number of
literals in any clause of the proof. In this paper we relate proof width
to proof length (=size), in both general Resolution, and its tree-like
variant. The following consequences of these relations reveal width as ... more >>>

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