Suppose Alice and Bob are communicating bits to each other in order to compute some function $f$, but instead of a classical communication channel they have a pair of walkie-talkie devices. They can use some classical communication protocol for $f$ where each round one player sends bit and the other ... more >>>

Half-duplex communication complexity with adversary was defined in [Hoover, K., Impagliazzo, R., Mihajlin, I., Smal, A. V. Half-Duplex Communication Complexity, ISAAC 2018.] Half-duplex communication protocols generalize classical protocols defined by Andrew Yao in [Yao, A. C.-C. Some Complexity Questions Related to Distributive Computing (Preliminary Report), STOC 1979]. It has been ... more >>>

Assuming that the class TAUT of tautologies of propositional logic has no almost optimal algorithm, we show that every algorithm $\mathbb A$ deciding TAUT has a polynomial time computable sequence witnessing that $\mathbb A$ is not almost optimal. The result extends to every $\Pi_t^p$-complete problem with $t\ge 1$; however, we ... more >>>

The existence of "unstructured" hard languages in $\text{NP}\cap\text{coNP}$ is an intriguing open question. Bennett and Gill (SICOMP, 1981) asked whether $\text{P}$ is separated from $\text{NP}\cap\text{coNP}$ relative to a random oracle, a question that remained open ever since. While a hard language in $\text{NP}\cap\text{coNP}$ can be constructed in a black-box way ... more >>>

We prove the first proof size lower bounds for the proof system Merge Resolution (MRes [Olaf Beyersdorff et al., 2020]), a refutational proof system for prenex quantified Boolean formulas (QBF) with a CNF matrix. Unlike most QBF resolution systems in the literature, proofs in MRes consist of resolution steps together ... more >>>

Itsykson and Sokolov in 2014 introduced the class of DPLL($\oplus$) algorithms that solve Boolean satisfiability problem using the splitting by linear combinations of variables modulo 2. This class extends the class of DPLL algorithms that split by variables. DPLL($\oplus$) algorithms solve in polynomial time systems of linear equations modulo two ... more >>>

Given a non-negative real matrix $M$ of non-negative rank at least $r$, can we witness this fact by a small submatrix of $M$? While Moitra (SIAM J. Comput. 2013) proved that this cannot be achieved exactly, we show that such a witnessing is possible approximately: an $m\times n$ matrix always ... more >>>

In a recent work, Gryaznov, Pudlák and Talebanfard (CCC '22) introduced a linear variant of read-once
branching programs, with motivations from circuit and proof complexity. Such a read-once linear branching program is
a branching program where each node is allowed to make $\mathbb{F}_2$-linear queries, and are read-once in the ... more >>>

We establish a generic form of hardness amplification for the approximability of constant-depth Boolean circuits by polynomials. Specifically, we show that if a Boolean circuit cannot be pointwise approximated by low-degree polynomials to within constant error in a certain one-sided sense, then an OR of disjoint copies of that circuit ... more >>>

An errorless heuristic is an algorithm that on all inputs returns either the correct answer or the special symbol "I don't know." A central question in average-case complexity is whether every distributional decision problem in NP has an errorless heuristic scheme: This is an algorithm that, for every δ > ... more >>>

We show that proving mildly super-linear lower bounds on non-commutative arithmetic circuits implies exponential lower bounds on non-commutative circuits. That is, non-commutative circuit complexity is a threshold phenomenon: an apparently weak lower bound actually suffices to show the strongest lower bounds we could desire.

This is part of a recent ... more >>>

We present a generic method for converting any family of unsatisfiable CNF formulas that require large resolution rank into CNF formulas whose refutation requires large rank for proof systems that manipulate polynomials or polynomial threshold functions of degree at most $k$ (known as ${\rm Th}(k)$ proofs). Such systems include: Lovasz-Schrijver ... more >>>

In this paper, we prove a general hardness amplification scheme for optimization problems based on the technique of direct products.

We say that an optimization problem $\Pi$ is direct product feasible if it is possible to efficiently aggregate any $k$ instances of $\Pi$ and form one large instance ... more >>>

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 >>>

We prove a version of the derandomized Direct Product Lemma for
deterministic space-bounded algorithms. Suppose a Boolean function
$g:\{0,1\}^n\to\{0,1\}$ cannot be computed on more than $1-\delta$
fraction of inputs by any deterministic time $T$ and space $S$
algorithm, where $\delta\leq 1/t$ for some $t$. Then, for $t$-step
walks $w=(v_1,\dots, v_t)$ ... more >>>

We systematically study the hardness and the approximability of combinatorial multi-objective NP optimization problems (multi-objective problems, for short).

We define solution notions that precisely capture the typical algorithmic tasks in multi-objective optimization. These notions inherit polynomial-time Turing reducibility from multivalued functions, which allows us to compare the solution notions and ... more >>>

We prove the computational hardness of three k-clustering problems using an (almost) arbitrary Bregman divergence as dissimilarity measure: (a) The Bregman k-center problem, where the objective is to find a set of centers that minimizes the maximum dissimilarity of any input point towards its closest center, and (b) the Bregman ... more >>>

Quantified Boolean Formulas (QBFs) extend propositional formulas with Boolean quantifiers. Working with QBF differs from propositional logic in its quantifier handling, but as propositional satisfiability (SAT) is a subproblem of QBF, all SAT hardness in solving and proof complexity transfers to QBF. This makes it difficult to analyse efforts dealing ... more >>>

We provide a tight characterisation of proof size in resolution for quantified Boolean formulas (QBF) by circuit complexity. Such a characterisation was previously obtained for a hierarchy of QBF Frege systems (Beyersdorff & Pich, LICS 2016), but leaving open the most important case of QBF resolution. Different from the Frege ... more >>>

Can every $n$-bit boolean function with deterministic query complexity $k\ll n$ be restricted to $O(k)$ variables such that the query complexity remains $\Omega(k)$? That is, can query complexity be condensed via restriction? We study such hardness condensation questions in both query and communication complexity, proving two main results.

$\bullet~$ $\mathbf{Negative}$: ... more >>>

We consider hypotheses about nondeterministic computation that
have been studied in different contexts and shown to have interesting
consequences:

1. The measure hypothesis: NP does not have p-measure 0.

2. The pseudo-NP hypothesis: there is an NP language that can be
distinguished from any DTIME(2^n^epsilon) language by an ... more >>>

In the Minimum Circuit Size Problem (MCSP[s(m)]), we ask if there is a circuit of size s(m) computing a given truth-table of length n = 2^m. Recently, a surprising phenomenon termed as hardness magnification by [Oliveira and Santhanam, FOCS 2018] was discovered for MCSP[s(m)] and the related problem MKtP of ... more >>>

We show that for several natural problems of interest, complexity lower bounds that are barely non-trivial imply super-polynomial or even exponential lower bounds in strong computational models. We term this phenomenon "hardness magnification". Our examples of hardness magnification include:

1. Let MCSP$[s]$ be the decision problem whose YES instances are ... more >>>

This work continues the development of hardness magnification. The latter proposes a strategy for showing strong complexity lower bounds by reducing them to a refined analysis of weaker models, where combinatorial techniques might be successful.

We consider gap versions of the meta-computational problems MKtP and MCSP, where one needs ... more >>>

Consider a communication network represented by a directed graph $G=(V,E)$ of $n$ nodes and $m$ edges. Assume that edges in $E$ are
partitioned into two sets: a set $C$ of edges with a fixed
non-negative real cost, and a set $P$ of edges whose \emph{price} should instead be set by ... more >>>

We introduce the notion of covering complexity of a probabilistic
verifier. The covering complexity of a verifier on a given input is
the minimum number of proofs needed to ``satisfy'' the verifier on
every random string, i.e., on every random string, at least one of the
given proofs must be ... more >>>

Producing a small DNF expression consistent with given data is a
classical problem in computer science that occurs in a number of forms and
has numerous applications. We consider two standard variants of this
problem. The first one is two-level logic minimization or finding a minimal
more >>>

In this paper, we consider the problem of approximately solving a
system of homogeneous linear equations over reals, where each
equation contains at most three variables.

Since the all-zero assignment always satisfies all the equations
exactly, we restrict the assignments to be ``non-trivial". Here is
an informal statement of our ... more >>>

We show that the minimum distance of a linear code (or
equivalently, the weight of the lightest codeword) is
not approximable to within any constant factor in random polynomial
time (RP), unless NP equals RP.
Under the stronger assumption that NP is not contained in RQP
(random ... more >>>

Recently Bansal and Sviridenko (Proc. of the 15th SODA'04, 189-196)
proved that for
2-dimensional Orthogonal Rectangle
Bin Packing without rotations allowed there is no asymptotic PTAS, unless P=NP. We show that similar
approximation hardness results hold for several rectangle packing and covering problems even if rotations by ninety
more >>>

We show that it is quasi-NP-hard to color $2$-colorable $12$-uniform hypergraphs with $2^{(\log n)^{\Omega(1) }}$ colors where $n$ is the number of vertices. Previously, Guruswami et al. [GHHSV14] showed that it is quasi-NP-hard to color $2$-colorable $8$-uniform hypergraphs with $2^{2^{\Omega(\sqrt{\log \log n})}}$ colors. Their result is obtained by composing a ... more >>>

How difficult is it to compute the communication complexity of a two-argument total Boolean function $f:[N]\times[N]\to\{0,1\}$, when it is given as an $N\times N$ binary matrix? In 2009, Kushilevitz and Weinreb showed that this problem is cryptographically hard, but it is still open whether it is NP-hard.

In this ... more >>>

Local search proved to be an extremely useful tool when facing hard optimization problems (e.g. via the simplex algorithm, simulated annealing, or genetic algorithms). Although powerful, it has its limitations: there are functions for which exponentially many queries are needed to find a local optimum. In many contexts the optimization ... more >>>

Given a graph G and a collection of source-sink pairs in G, what is the least integer c such that each source can be connected by a path to its sink, with at most c paths going through an edge? This is known as the congestion minimization problem, and the ... more >>>

We consider several closely related variants of PAC-learning in which false-positive and false-negative errors are treated differently. In these models we seek to guarantee a given, low rate of false-positive errors and as few false-negative errors as possible given that we meet the false-positive constraint. Bshouty and Burroughs first observed ... more >>>

A recent breakthrough of Liu and Pass (FOCS'20) shows that one-way functions exist if and only if the (polynomial-)time-bounded Kolmogorov complexity K^t is bounded-error hard on average to compute. In this paper, we strengthen this result and extend it to other complexity measures:

1. We show, perhaps surprisingly, that the ... more >>>

The problem of learning $t$-term DNF formulas (for $t = O(1)$) has been studied extensively in the PAC model since its introduction by Valiant (STOC 1984). A $t$-term DNF can be efficiently learnt using a $t$-term DNF only if $t = 1$ i.e., when it is an AND, while even ... more >>>

Learning an unknown halfspace (also called a perceptron) from
labeled examples is one of the classic problems in machine learning.
In the noise-free case, when a halfspace consistent with all the
training examples exists, the problem can be solved in polynomial
time using linear programming. ... more >>>

We prove the hardness of weakly learning halfspaces in the presence of adversarial noise using polynomial threshold functions (PTFs). In particular, we prove that for any constants $d \in \mathbb{Z}^+$ and $\epsilon > 0$, it is NP-hard to decide: given a set of $\{-1,1\}$-labeled points in $\mathbb{R}^n$ whether (YES Case) ... more >>>

We prove a strong inapproximability result for routing on directed
graphs with low congestion. Given as input a directed graph on $N$
vertices and a set of source-destination pairs that can be connected
via edge-disjoint paths, we prove that it is hard, assuming NP
doesn't have $n^{O(\log\log n)}$ time randomized ... more >>>

Determinantal Point Processes (DPPs) are a widely used probabilistic model for negatively correlated sets. DPPs have been successfully employed in Machine Learning applications to select a diverse, yet representative subset of data. In these applications, the parameters of the DPP need to be fitted to match the data; typically, we ... more >>>

Parameterized Resolution and, moreover, a general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider (FOCS'07). In that paper, Dantchev et al. show a complexity gap in tree-like Parameterized Resolution for propositional formulas arising from translations of first-order principles.
We broadly investigate Parameterized Resolution obtaining the following ... more >>>

A hypergraph is $k$-rainbow colorable if there exists a vertex coloring using $k$ colors such that each hyperedge has all the $k$ colors. Unlike usual hypergraph coloring, rainbow coloring becomes harder as the number of colors increases. This work studies the rainbow colorability of hypergraphs which are guaranteed to be ... more >>>

A recent line of research has introduced a systematic approach to explore the complexity of explicit construction problems through the use of meta problems, namely, the range avoidance problem (abbrev. Avoid) and the remote point problem (abbrev. RPP). The upper and lower bounds for these meta problems provide a unified ... more >>>

We study the polynomial reconstruction problem for low-degree
multivariate polynomials over finite fields. In the GF[2] version of this problem, we are given a set of points on the hypercube and target values $f(x)$ for each of these points, with the promise that there is a polynomial over GF[2] of ... more >>>

A classic result due to Hastad established that for every constant \eps > 0, given an overdetermined system of linear equations over a finite field \F_q where each equation depends on exactly 3 variables and at least a fraction (1-\eps) of the equations can be satisfied, it is NP-hard to ... more >>>

We show that one-way functions exist if and only if there is some samplable distribution D such that it is hard to approximate the Kolmogorov complexity of a string sampled from D. Thus we characterize the existence of one-way functions by the average-case hardness of a natural \emph{uncomputable} problem on ... more >>>

A common method for increasing the usability and uplifting the security of pseudorandom function families (PRFs) is to ``hash" the inputs into a smaller domain before applying the PRF. This approach, known as ``Levin's trick", is used to achieve ``PRF domain extension" (using a short, e.g., fixed, input length PRF ... more >>>

We consider the question of hardness self-amplification: Given a Boolean function $f$ that is hard to compute on a $o(1)$-fraction of inputs drawn from some distribution, can we prove that $f$ is hard to compute on a $(\frac{1}{2} - o(1))$-fraction of inputs drawn from the same distribution? We prove hardness ... more >>>

Strong (resp. weak) average-case hardness refers to the properties of a computational problem in which a large (resp. small) fraction of instances are hard to solve. We develop a general framework for proving hardness self-amplification, that is, the equivalence between strong and weak average-case hardness. Using this framework, we prove ... more >>>

We propose a new approach to the hardness-to-randomness framework and to the promise-BPP=promise-P conjecture. Classical results rely on non-uniform hardness assumptions to construct derandomization algorithms that work in the worst-case, or rely on uniform hardness assumptions to construct derandomization algorithms that work only in the average-case. In both types of ... more >>>

In this paper we show that lower bounds for bounded depth arithmetic circuits imply derandomization of polynomial identity testing for bounded depth arithmetic circuits. More formally, if there exists an explicit polynomial f(x_1,...,x_m) that cannot be computed by a depth d arithmetic circuit of small size then there exists ... more >>>

Bennett and Gill (1981) proved that P^A != NP^A relative to a
random oracle A, or in other words, that the set
O_[P=NP] = { A | P^A = NP^A }
has Lebesgue measure 0. In contrast, we show that O_[P=NP] has
Hausdorff dimension 1.

... more >>>

More than twenty years ago, Capalbo, Reingold, Vadhan and Wigderson gave the first (and up to date only) explicit construction of a bipartite expander with almost full combinatorial expansion. The construction incorporates zig-zag ideas together with extractor technology, and is rather complicated. We give an alternative construction that builds upon ... more >>>

Height restricted constant depth LK is a natural restriction of
Gentzen's propositional proof system LK. A sequence of LK-formulas
has polylogarithmic-height restricted depth-d-LK proofs iff the
n-th formula in the sequence possesses LK-proofs where cut-formulas
are of depth d+1 with small bottom fanin
and of ... more >>>

We present logspace algorithms for the canonical labeling problem and the representation problem of Helly circular-arc (HCA) graphs. The first step is a reduction to canonical labeling and representation of interval intersection matrices. In a second step, the Delta trees employed in McConnell's linear time representation algorithm for interval matrices ... more >>>

We give a new simple proof of the time hierarchy theorem for heuristic BPP originally proved by Fortnow and Santhanam [FS04] and then simplified and improved by Pervyshev [P07]. In the proof we use a hierarchy theorem for sampling distributions recently proved by Watson [W13]. As a byproduct we get ... more >>>

We strengthen the non-deterministic time hierarchy theorem of
\cite{Cook72, Seiferas-Fischer-Meyer78, Zak83} to show that the lower bound
holds against sublinear advice. More formally, we show that for any constants
$c$ and $d$ such that $1 \leq c < d$, there is a language in $\NTIME(n^d)$
which is not in $\NTIME(n^c)/n^{1/d}$. ... more >>>

Referring to the query complexity of property testing,
we prove the existence of a rich hierarchy of corresponding
complexity classes. That is, for any relevant function $q$,
we prove the existence of properties that have testing
complexity Theta(q).
Such results are proven in three standard
domains often considered in property ... more >>>

Focusing on property testing tasks that have query complexity that is independent of the size of the tested object (i.e., depends on the proximity parameter only), we prove the existence of a rich hierarchy of the corresponding complexity classes.
That is, for essentially any function $q:(0,1]\to\N$, we prove the existence ... more >>>

Chvatal-Gomory (CG) cuts and the Bienstock-Zuckerberg hierarchy capture useful linear programs that the standard bounded degree Lasserre/Sum-of-Squares (SOS) hierarchy fails to capture.

In this paper we present a novel polynomial time SOS hierarchy for 0/1 problems with a custom subspace of high degree polynomials (not the standard subspace of low-degree ... more >>>

We show that high dimensional expanders imply derandomized direct product tests, with a number of subsets that is *linear* in the size of the universe.

Direct product tests belong to a family of tests called agreement tests that are important components in PCP constructions and include, for example, low degree ... more >>>

Higher order random walks (HD-walks) on high dimensional expanders have played a crucial role in a number of recent breakthroughs in theoretical computer science, perhaps most famously in the recent resolution of the Mihail-Vazirani conjecture (Anari et al. STOC 2019), which focuses on HD-walks on one-sided local-spectral expanders. In this ... more >>>

We present a general framework for constructing high rate error correcting codes that are locally correctable (and hence locally decodable if linear) with a sublinear number of queries, based on lifting codes with respect to functions on the coordinates. Our approach generalizes the lifting of affine-invariant codes of Guo, Kopparty, ... more >>>

In this work, we construct the first locally-correctable codes (LCCs), and locally-testable codes (LTCs) with constant rate, constant relative distance, and sub-polynomial query complexity. Specifically, we show that there exist binary LCCs and LTCs with block length $n$, constant rate (which can even be taken arbitrarily close to 1), constant ... more >>>

The classical Reed-Muller codes over a finite field $\mathbb{F}_q$ are based on evaluations of $m$-variate polynomials of degree at most $d$ over a product set $U^m$, for some $d$ less than $|U|$. Because of their good distance properties, as well as the ubiquity and expressive power of polynomials, these codes ... more >>>

An error correcting code $\mathcal{C} \colon \Sigma^k \to \Sigma^n$ is list-recoverable from input list size $\ell$ if for any sets $\mathcal{L}_1, \ldots, \mathcal{L}_n \subseteq \Sigma$ of size at most $\ell$, one can efficiently recover the list $\mathcal{L} = \{ x \in \Sigma^k : \forall j \in [n], \mathcal{C}(x)_j \in \mathcal{L}_j ... more >>>

Locally decodable codes are error-correcting codes that admit efficient decoding algorithms; any bit of the original message can be recovered by looking at only a small number of locations of a corrupted codeword. The tradeoff between the rate of a code and the locality/efficiency of its decoding algorithms has been ... more >>>

An error correcting code is said to be \emph{locally testable} if
there is a test that checks whether a given string is a codeword,
or rather far from the code, by reading only a small number of symbols
of the string. Locally testable codes (LTCs) are both interesting
in their ... more >>>

In this article we are interested in the density of small linear structures (e.g. arithmetic progressions) in subsets $A$ of the group $\mathbb{F}_p^n$. It is possible to express these densities as certain analytic averages involving $1_A$, the indicator function of $A$. In the higher-order Fourier analytic approach, the function $1_A$ ... more >>>

For $S\subseteq \mathbb{F}^n$, consider the linear space of restrictions of degree-$d$ polynomials to $S$. The Hilbert function of $S$, denoted $\mathrm{h}_S(d,\mathbb{F})$, is the dimension of this space. We obtain a tight lower bound on the smallest value of the Hilbert function of subsets $S$ of arbitrary finite grids in $\mathbb{F}^n$ ... more >>>

We present the first worst-case hardness conditions
on the circuit complexity of EXP functions which are
sufficient to obtain P=BPP. In particular, we show that
from such hardness conditions it is possible to construct
quick Hitting Sets Generators with logarithmic prize.
... more >>>

In this paper we study polynomials in VP$_e$ (polynomial-sized formulas) and in $\Sigma\Pi\Sigma$ (polynomial-size depth-$3$ circuits) whose orbits, under the action of the affine group GL$^{aff}_n({\mathbb F})$, are dense in their ambient class. We construct hitting sets and interpolating sets for these orbits as well as give reconstruction algorithms.

As ... more >>>

We show that hitting sets can derandomize any BPP-algorithm.
This gives a positive answer to a fundamental open question in
probabilistic algorithms. More precisely, we present a polynomial
time deterministic algorithm which uses any given hitting set
to approximate the fractions of 1's in the ... more >>>

We give a length-efficient puncturing of Reed-Muller codes which preserves its distance properties. Formally, for the Reed-Muller code encoding $n$-variate degree-$d$ polynomials over ${\mathbb F}_q$ with $q \ge \Omega(d/\delta)$, we present an explicit (multi)-set $S \subseteq {\mathbb F}_q^n$ of size $N=\mathrm{poly}(n^d/\delta)$ such that every nonzero polynomial vanishes on at most ... more >>>

The orbit of an $n$-variate polynomial $f(\mathbf{x})$ over a field $\mathbb{F}$ is the set $\mathrm{orb}(f) := \{f(A\mathbf{x}+\mathbf{b}) : A \in \mathrm{GL}(n,\mathbb{F}) \ \mathrm{and} \ \mathbf{b} \in \mathbb{F}^n\}$. This paper studies explicit hitting sets for the orbits of polynomials computable by certain well-studied circuit classes. This version of the hitting set ... more >>>

We construct an explicit $\varepsilon$-hitting set generator (HSG) for regular ordered branching programs of length $n$ and $\textit{unbounded width}$ with a single accept state that has seed length
\[O(\log(n)(\log\log(n)+\log(1/\varepsilon))).\]
Previously, the best known seed length for regular branching programs of width $w$ with a single accept state was ... more >>>

A hitting set is a "one-sided" variant of a pseudorandom generator (PRG), naturally suited to derandomizing algorithms that have one-sided error. We study the problem of using a given hitting set to derandomize algorithms that have two-sided error, focusing on space-bounded algorithms. For our first result, we show that if ... more >>>

Nisan (Combinatorica'92) constructed a pseudorandom generator for length $n$, width $n$ read-once branching programs (ROBPs) with error $\varepsilon$ and seed length $O(\log^2{n} + \log{n} \cdot \log(1/\varepsilon))$. A major goal in complexity theory is to reduce the seed length, hopefully, to the optimal $O(\log{n}+\log(1/\varepsilon))$, or to construct improved hitting sets, as ... more >>>

The depth-$3$ model has recently gained much importance, as it has become a stepping-stone to understanding general arithmetic circuits. Its restriction to multilinearity has known exponential lower bounds but no nontrivial blackbox identity tests. In this paper we take a step towards designing such hitting-sets. We define a notion of ... more >>>

We give a $n^{O(\log n)}$-time ($n$ is the input size) blackbox polynomial identity testing algorithm for unknown-order read-once oblivious algebraic branching programs (ROABP). The best time-complexity known for this class was $n^{O(\log^2 n)}$ due to Forbes-Saptharishi-Shpilka (STOC 2014), and that too only for multilinear ROABP. We get rid of their ... more >>>

Complexity theory is built fundamentally on the notion of efficient
reduction among computational problems. Classical
reductions involve gadgets that map solution fragments of one problem to
solution fragments of another in one-to-one, or
possibly one-to-many, fashion. In this paper we propose a new kind of
reduction that allows for gadgets ... more >>>

We develop the theory of holographic algorithms. We give
characterizations of algebraic varieties of realizable
symmetric generators and recognizers on the basis manifold,
and a polynomial time decision algorithm for the
simultaneous realizability problem.
Using the general machinery we are able to give
unexpected holographic algorithms for
some counting problems, ... more >>>

Valiant's theory of holographic algorithms is a novel methodology
to achieve exponential speed-ups in computation. A fundamental
parameter in holographic algorithms is the dimension of the linear basis
vectors.
We completely resolve the problem of the power of higher dimensional
bases. We prove that 2-dimensional bases are universal for
holographic ... more >>>

We show that any private-key encryption scheme that is weakly
homomorphic with respect to addition modulo 2, can be transformed
into a public-key encryption scheme. The homomorphic feature
referred to is a minimalistic one; that is, the length of a
homomorphically generated encryption should be independent of the
number of ... more >>>

The VP versus VNP question, introduced by Valiant, is probably the most important open question in algebraic complexity theory. Thanks to completeness results, a variant of this question, VBP versus VNP, can be succinctly restated as asking whether the permanent of a generic matrix can be written as a determinant ... more >>>

Property-testers are fast randomized algorithms for distinguishing
between graphs (and other combinatorial structures) satisfying a
certain property, from those that are far from satisfying it. In
many cases one can design property-testers whose running time is in
fact {\em independent} of the size of the input. In this paper we
more >>>

We obtain the first true size-space trade-offs for the cutting planes proof system, where the upper bounds hold for size and total space for derivations with constant-size coefficients, and the lower bounds apply to length and formula space (i.e., number of inequalities in memory) even for derivations with exponentially large ... more >>>

Let $f$ be a non-commutative polynomial such that $f=0$ if we assume that the variables in $f$ commute. Let $Q(f)$ be the smallest $k$ such that there exist polynomials $g_1,g_1', g_2, g_2',\dots, g_k, g_k' $ with \[f\in I([g_1,g_1'], [g_2, g_2'],\dots, [g_k, g_k'] )\,,\]
where $[g,h]=gh-hg$. Then $Q(f)\leq {n\choose 2}$, where ... more >>>

The Merlin-Arthur class of languages MA is included into Arthur-Merlin class AM, and into PP. For a standard transformation of a given MA protocol with Arthur's message (= random string) of length $a$ and Merlin's message of length $m$ to a PP machine, the latter needs $O(ma)$ random bits. The ... more >>>

The question whether Identity-Based Encryption (IBE) can be based on the Decisional Diffie-Hellman (DDH) assumption is one of the most prominent questions in Cryptography related to DDH. We study limitations on the use of the DDH assumption in cryptographic constructions, and show that it is impossible to construct a secure ... more >>>

In this paper we show that the QBF proof checking format QRAT (Quantified Resolution Asymmetric Tautologies) by Heule, Biere and Seidl cannot have polynomial-time strategy extraction unless P=PSPACE. In our proof, the crucial property that makes strategy extraction PSPACE-hard for this proof format is universal expansion, even expansion on a ... more >>>

Relaxations for the constraint satisfaction problem (CSP) include bounded width, linear program (LP), semidefinite program (SDP), afinfe integer program (AIP), and the combined LP+AIP of Brakensiek, Guruswami, Wrochna, and Živný (SICOMP 2020). Tightening relaxations systematically leads to hierarchies and stronger algorithms. For the LP+AIP hierarchy, a constant level lower bound ... more >>>

Despite over 25 years of research on non-malleable commitments in the plain model, their round complexity has remained open. The goal of achieving non-malleable commitment protocols with only one or two rounds has been especially elusive. Pass (TCC 2013, CC 2016) captured this difficulty by proving important impossibility results regarding ... more >>>

Perfect zero-knowledge (PZK) proofs have been constructed in various settings and they are
also interesting from a complexity theoretic perspective. Yet, virtually nothing is known about them. This is in contrast to statistical zero-knowledge proofs, where many general results have been proved using an array of tools, none of which ... more >>>

We study the relationship between communication and information in 2-party communication protocols when the information is asymmetric. If $I^A$ denotes the number of bits of information revealed by the first party, $I^B$ denotes the information revealed by the second party, and $C$ is the number of bits of communication in ... more >>>

We address the following problem: how to execute any algorithm P, for an unbounded number of executions, in the presence of an adversary who observes partial information on the internal state of the computation during executions. The security guarantee is that the adversary learns nothing, beyond P's input/output behavior.

This ... more >>>

In the presence of a quantum adversary, there are two possible definitions of security for a pseudorandom function. The first, which we call standard-security, allows the adversary to be quantum, but requires queries to the function to be classical. The second, quantum-security, allows the adversary to query the function on ... more >>>

We construct a 1-round delegation scheme (i.e., argument-system) for every language computable in time t=t(n), where the running time of the prover is poly(t) and the running time of the verifier is n*polylog(t). In particular, for every language in P we obtain a delegation scheme with almost linear time verification. ... more >>>

The focus of this survey is the question of *quantified derandomization*, which was introduced by Goldreich and Wigderson (2014): Does derandomization of probabilistic algorithms become easier if we only want to derandomize algorithms that err with extremely small probability? How small does this probability need to be in order for ... more >>>

Yao's garbled circuit construction transforms a boolean circuit $C:\{0,1\}^n\to\{0,1\}^m$
into a ``garbled circuit'' $\hat{C}$ along with $n$ pairs of $k$-bit keys, one for each
input bit, such that $\hat{C}$ together with the $n$ keys
corresponding to an input $x$ reveal $C(x)$ and no additional information about $x$.
The garbled circuit ... 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 construct a secure protocol for any multi-party functionality
that remains secure (under a relaxed definition of security) when
executed concurrently with multiple copies of itself and other
protocols. We stress that we do *not* use any assumptions on
existence of trusted parties, common reference string, honest
majority or synchronicity ... more >>>

In this note we improve a recent result by Arora, Khot, Kolla, Steurer, Tulsiani, and Vishnoi on solving the Unique Games problem on expanders.

Given a (1 - epsilon)-satisfiable instance of Unique Games with the constraint graph G, our algorithm finds an assignments satisfying at least a (1 - C ... more >>>

Suppose you ran a chess tournament, everybody played everybody, and you wanted to use the results to rank everybody. Unless you were really lucky, the results would not be acyclic, so you could not just sort the players by who beat whom. A natural objective is to find a ranking ... more >>>

Threshold cryptography is typically based on the idea of secret-sharing a private-key $s\in F$ ``in the exponent'' of some cryptographic group $G$, or more generally, encoding $s$ in some linearly homomorphic domain. In each invocation of the threshold system (e.g., for signing or decrypting) an ``encoding'' of the secret is ... more >>>

Secret sharing schemes allow a dealer to distribute a secret piece of information among several parties so that any qualified subset of parties can reconstruct the secret, while every unqualified subset of parties learns nothing about the secret. The collection of qualified subsets is called an access structure. The best ... more >>>

One of the most fundamental notions of cryptography is that of \emph{simulation}. It stands behind the concepts of semantic security, zero knowledge, and security for multiparty computation. However, writing a simulator and proving security via the use of simulation is a non-trivial task, and one that many newcomers to the ... more >>>

Let $p(x_1,...,x_n) = p(X) , X \in R^{n}$ be a homogeneous polynomial of degree $n$ in $n$ real variables ,
$e = (1,1,..,1) \in R^n$ be a vector of all ones . Such polynomial $p$ is
called $e$-hyperbolic if for all real vectors $X \in R^{n}$ the univariate polynomial
equation ... more >>>

Hypercontractivity is one of the most powerful tools in Boolean function analysis. Originally studied over the discrete hypercube, recent years have seen increasing interest in extensions to settings like the $p$-biased cube, slice, or Grassmannian, where variants of hypercontractivity have found a number of breakthrough applications including the resolution of ... more >>>

We prove hypercontractive inequalities on high dimensional expanders. As in the settings of the p-biased hypercube, the symmetric group, and the Grassmann scheme, our inequalities are effective for global functions, which are functions that are not significantly affected by a restriction of a small set of coordinates. As applications, we ... more >>>