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Electronic Colloquium on Computational Complexity

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Reports tagged with XOR Lemmas:
TR10-072 | 19th April 2010
Russell Impagliazzo, Valentine Kabanets

Constructive Proofs of Concentration Bounds

We give a simple combinatorial proof of the Chernoff-Hoeffding concentration bound~\cite{Chernoff, Hof63}, which says that the sum of independent $\{0,1\}$-valued random variables is highly concentrated around the expected value. Unlike the standard proofs,
our proof does not use the method of higher moments, but rather uses a simple ... more >>>

TR11-040 | 22nd March 2011
Alexander A. Sherstov

Strong Direct Product Theorems for Quantum Communication and Query Complexity

A strong direct product theorem (SDPT) states that solving $n$ instances of a problem requires $\Omega(n)$ times the resources for a single instance, even to achieve success probability $2^{-\Omega(n)}.$ We prove that quantum communication complexity obeys an SDPT whenever the communication lower bound for a single instance is proved by ... more >>>

TR11-145 | 2nd November 2011
Alexander A. Sherstov

The Multiparty Communication Complexity of Set Disjointness

We study the set disjointness problem in the number-on-the-forehead model.

(i) We prove that $k$-party set disjointness has randomized and nondeterministic
communication complexity $\Omega(n/4^k)^{1/4}$ and Merlin-Arthur complexity $\Omega(n/4^k)^{1/8}.$
These bounds are close to tight. Previous lower bounds (2007-2008) for $k\geq3$ parties
were weaker than $n^{1/(k+1)}/2^{k^2}$ in all ... more >>>

TR18-085 | 26th April 2018
Andrej Bogdanov, Manuel Sabin, Prashant Nalini Vasudevan

XOR Codes and Sparse Random Linear Equations with Noise

A $k$-LIN instance is a system of $m$ equations over $n$ variables of the form $s_{i[1]} + \dots + s_{i[k]} =$ 0 or 1 modulo 2 (each involving $k$ variables). We consider two distributions on instances in which the variables are chosen independently and uniformly but the right-hand sides are ... more >>>

TR19-105 | 16th August 2019
Ragesh Jaiswal

A note on the relation between XOR and Selective XOR Lemmas

Given an unpredictable Boolean function $f: \{0, 1\}^n \rightarrow \{0, 1\}$, the standard Yao's XOR lemma is a statement about the unpredictability of computing $\oplus_{i \in [k]}f(x_i)$ given $x_1, ..., x_k \in \{0, 1\}^n$, whereas the Selective XOR lemma is a statement about the unpredictability of computing $\oplus_{i \in S}f(x_i)$ ... more >>>

TR19-145 | 31st October 2019
Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, Shachar Lovett, David Zuckerman

XOR Lemmas for Resilient Functions Against Polynomials

A major challenge in complexity theory is to explicitly construct functions that have small correlation with low-degree polynomials over $F_2$. We introduce a new technique to prove such correlation bounds with $F_2$ polynomials. Using this technique, we bound the correlation of an XOR of Majorities with constant degree polynomials. In ... more >>>

TR20-124 | 3rd August 2020
Joshua Brody, JaeTak Kim, Peem Lerdputtipongporn, Hariharan Srinivasulu

A Strong XOR Lemma for Randomized Query Complexity

We give a strong direct sum theorem for computing $XOR \circ g$. Specifically, we show that the randomized query complexity of computing the XOR of $k$ instances of $g$ satisfies $\bar{R}_\varepsilon(XOR \circ g)=\Theta(\bar{R}_{\varepsilon/k}(g))$. This matches the naive success amplification bound and answers a question of Blais and Brody.

As a ... more >>>

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