Arkadev Chattopadhyay

Let m,q > 1 be two integers that are co-prime and A be any subset of Z_m. Let P be any multi-linear polynomial of degree d in n variables over Z_m. We show that the MOD_q boolean function on n variables has correlation at most exp(-\Omega(n/(m2^{m-1})^d)) with the boolean function ... more >>>

Scott Aaronson, Avi Wigderson

Any proof of P!=NP will have to overcome two barriers: relativization

and natural proofs. Yet over the last decade, we have seen circuit

lower bounds (for example, that PP does not have linear-size circuits)

that overcome both barriers simultaneously. So the question arises of

whether there ...
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Shachar Lovett

We study the density of the weights of Generalized Reed--Muller codes. Let $RM_p(r,m)$ denote the code of multivariate polynomials over $\F_p$ in $m$ variables of total degree at most $r$. We consider the case of fixed degree $r$, when we let the number of variables $m$ tend to infinity. We ... more >>>

Arnab Bhattacharyya, Swastik Kopparty, Grant Schoenebeck, Madhu Sudan, David Zuckerman

We consider the problem of testing if a given function

$f : \F_2^n \rightarrow \F_2$ is close to any degree $d$ polynomial

in $n$ variables, also known as the Reed-Muller testing problem.

Alon et al.~\cite{AKKLR} proposed and analyzed a natural

$2^{d+1}$-query test for this property and showed that it accepts

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Shachar Lovett, Partha Mukhopadhyay, Amir Shpilka

In this paper we give the first construction of a pseudorandom generator, with seed length $O(\log n)$, for $\mathrm{CC}_0[p]$, the class of constant-depth circuits with unbounded fan-in $\mathrm{MOD}_p$ gates, for some prime $p$. More accurately, the seed length of our generator is $O(\log n)$ for any constant error $\epsilon>0$. In ... more >>>

Albert Atserias, Anuj Dawar

Kolaitis and Kopparty have shown that for any first-order formula with

parity quantifiers over the language of graphs there is a family of

multi-variate polynomials of constant-degree that agree with the

formula on all but a $2^{-\Omega(n)}$-fraction of the graphs with $n$

vertices. The proof yields a bound on the ...
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Badih Ghazi, Pritish Kamath, Prasad Raghavendra

In this work we introduce a new technique for reducing the dimension of the ambient space of low-degree polynomials in the Gaussian space while preserving their relative correlation structure. As applications, we address the following problems:

(I) Computability of the Approximately Optimal Noise Stable function over Gaussian space:

The goal ... more >>>

Srikanth Srinivasan, Madhu Sudan

The well-known DeMillo-Lipton-Schwartz-Zippel lemma says that $n$-variate

polynomials of total degree at most $d$ over

grids, i.e. sets of the form $A_1 \times A_2 \times \cdots \times A_n$, form

error-correcting codes (of distance at least $2^{-d}$ provided $\min_i\{|A_i|\}\geq 2$).

In this work we explore their local

decodability and local testability. ...
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Marshall Ball, Eshan Chattopadhyay, Jyun-Jie Liao, Tal Malkin, Li-Yang Tan

We present the first explicit construction of a non-malleable code that can handle tampering functions that are bounded-degree polynomials.

Prior to our work, this was only known for degree-1 polynomials (affine tampering functions), due to Chattopadhyay and Li (STOC 2017). As a direct corollary, we obtain an explicit non-malleable ... more >>>