Igor E. Shparlinski

D. Boneh and R. Venkatesan have recently proposed an approach to proving

that a reasonably small portions of most significant bits of the

Diffie--Hellman key modulo a prime are as secure the the whole key. Some

further improvements and generalizations have been obtained by

I. M. Gonzales Vasco ...
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Zeev Dvir, Ariel Gabizon, Avi Wigderson

In this paper we construct explicit deterministic extractors from polynomial sources, namely from distributions sampled by low degree multivariate polynomials over finite fields. This naturally generalizes previous work on extraction from affine sources (which are degree 1 polynomials). A direct consequence is a deterministic extractor for distributions sampled by polynomial ... more >>>

Zeev Dvir, Swastik Kopparty, Shubhangi Saraf, Madhu Sudan

We extend the ``method of multiplicities'' to get the following results, of interest in combinatorics and randomness extraction.

\begin{enumerate}

\item We show that every Kakeya set in $\F_q^n$, the $n$-dimensional vector space over the finite field on $q$ elements, must be of size at least $q^n/2^n$. This bound is tight ...
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Parikshit Gopalan

We present a Fourier-analytic approach to list-decoding Reed-Muller codes over arbitrary finite fields. We prove that the list-decoding radius for quadratic polynomials equals $1 - 2/q$ over any field $F_q$ where $q > 2$. This confirms a conjecture due to Gopalan, Klivans and Zuckerman for degree $2$. Previously, tight bounds ... more >>>

Charanjit Jutla, Arnab Roy

We consider multivariate pseudo-linear functions

over finite fields of characteristic two. A pseudo-linear polynomial

is a sum of guarded linear-terms, where a guarded linear-term is a product of one or more linear-guards

and a single linear term, and each linear-guard is

again a linear term but raised ...
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Arnab Bhattacharyya, Zeev Dvir, Shubhangi Saraf, Amir Shpilka

A Locally Correctable Code (LCC) is an error correcting code that has a probabilistic

self-correcting algorithm that, with high probability, can correct any coordinate of the

codeword by looking at only a few other coordinates, even if a fraction $\delta$ of the

coordinates are corrupted. LCC's are a stronger form ...
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Johannes Mittmann, Nitin Saxena, Peter Scheiblechner

A set of multivariate polynomials, over a field of zero or large characteristic, can be tested for algebraic independence by the well-known Jacobian criterion. For fields of other characteristic $p>0$, there is no analogous characterization known. In this paper we give the first such criterion. Essentially, it boils down to ... more >>>

Swastik Kopparty

We study the list-decodability of multiplicity codes. These codes, which are based on evaluations of high-degree polynomials and their derivatives, have rate approaching $1$ while simultaneously allowing for sublinear-time error-correction. In this paper, we show that multiplicity codes also admit powerful list-decoding and local list-decoding algorithms correcting a large fraction ... more >>>

Eli Ben-Sasson, Noga Ron-Zewi, Madhu Sudan

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 >>>

Chaim Even-Zohar, Shachar Lovett

Let $G$ be a finite abelian group of torsion $r$ and let $A$ be a subset of $G$.

The Freiman-Ruzsa theorem asserts that if $|A+A| \le K|A|$

then $A$ is contained in a coset of a subgroup of $G$ of size at most $K^2 r^{K^4} |A|$. It was ...
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Arman Fazeli, Shachar Lovett, Alex Vardy

A $t$-$(n,k,\lambda)$ design over $\mathbb{F}_q$ is a collection of $k$-dimensional subspaces of $\mathbb{F}_q^n$, ($k$-subspaces, for short), called blocks, such that each $t$-dimensional subspace of $\mathbb{F}_q^n$ is contained in exactly $\lambda$ blocks. Such $t$-designs over $\mathbb{F}_q$ are the $q$-analogs of conventional combinatorial designs. Nontrivial $t$-$(n,k,\lambda)$ designs over $\mathbb{F}_q$ are currently known ... more >>>

Jean Bourgain, Zeev Dvir, Ethan Leeman

We describe a construction of explicit affine extractors over large finite fields with exponentially small error and linear output length. Our construction relies on a deep theorem of Deligne giving tight estimates for exponential sums over smooth varieties in high dimensions.

more >>>Arnab Bhattacharyya, Abhishek Bhowmick

Higher-order Fourier analysis, developed over prime fields, has been recently used in different areas of computer science, including list decoding, algorithmic decomposition and testing. We extend the tools of higher-order Fourier analysis to analyze functions over general fields. Using these new tools, we revisit the results in the above areas.

... more >>>Mrinal Kumar, Ramprasad Saptharishi

In this paper, we show exponential lower bounds for the class of homogeneous depth-$5$ circuits over all small finite fields. More formally, we show that there is an explicit family $\{P_d : d \in N\}$ of polynomials in $VNP$, where $P_d$ is of degree $d$ in $n = d^{O(1)}$ variables, ... more >>>

Shachar Lovett

The GM-MDS conjecture of Dau et al. (ISIT 2014) speculates that the MDS condition, which guarantees the existence of MDS matrices with a prescribed set of zeros over large fields, is in fact sufficient for existence of such matrices over small fields. We prove this conjecture.

Ankit Garg, Nikhil Gupta, Neeraj Kayal, Chandan Saha

The determinant polynomial $Det_n(\mathbf{x})$ of degree $n$ is the determinant of a $n \times n$ matrix of formal variables. A polynomial $f$ is equivalent to $Det_n$ over a field $\mathbf{F}$ if there exists a $A \in GL(n^2,\mathbf{F})$ such that $f = Det_n(A \cdot \mathbf{x})$. Determinant equivalence test over $\mathbf{F}$ is ... more >>>

Vishwas Bhargava, Sumanta Ghosh, Zeyu Guo, Mrinal Kumar, Chris Umans

Multivariate multipoint evaluation is the problem of evaluating a multivariate polynomial, given as a coefficient vector, simultaneously at multiple evaluation points. In this work, we show that there exists a deterministic algorithm for multivariate multipoint evaluation over any finite field $\mathbb{F}$ that outputs the evaluations of an $m$-variate polynomial of ... more >>>

Yogesh Dahiya, Meena Mahajan, Sasank Mouli

In this paper, we obtain new size lower bounds for proofs in the

Polynomial Calculus (PC) proof system, in two different settings.

1. When the Boolean variables are encoded using $\pm 1$ (as opposed

to $0,1$): We establish a lifting theorem using an asymmetric gadget

$G$, showing ...
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Kiran Kedlaya, Swastik Kopparty

For an odd prime $p$, we say $f(X) \in {\mathbb F}_p[X]$ computes square roots in $\mathbb F_p$ if, for all nonzero perfect squares $a \in \mathbb F_p$, we have $f(a)^2 = a$.

When $p \equiv 3$ mod $4$, it is well known that $f(X) = X^{(p+1)/4}$ computes square ...
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Shanthanu Rai

We present a polynomial-time pseudo-deterministic algorithm for constructing irreducible polynomial of degree $d$ over finite field $\mathbb{F}_q$. A pseudo-deterministic algorithm is allowed to use randomness, but with high probability it must output a canonical irreducible polynomial. Our construction runs in time $\tilde{O}(d^4 \log^4{q})$.

Our construction extends Shoup's deterministic algorithm ... more >>>