All reports by Author Mahdi Cheraghchi:

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TR22-156
| 15th November 2022
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Huck Bennett, Mahdi Cheraghchi, Venkatesan Guruswami, Joao Ribeiro#### Parameterized Inapproximability of the Minimum Distance Problem over all Fields and the Shortest Vector Problem in all $\ell_p$ Norms

Revisions: 1

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TR21-009
| 1st February 2021
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Eric Allender, Mahdi Cheraghchi, Dimitrios Myrisiotis, Harsha Tirumala, Ilya Volkovich#### One-way Functions and Partial MCSP

Revisions: 3
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Comments: 1

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TR20-103
| 9th July 2020
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Mahdi Cheraghchi, Shuichi Hirahara, Dimitrios Myrisiotis, Yuichi Yoshida#### One-Tape Turing Machine and Branching Program Lower Bounds for MCSP

Revisions: 1

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TR19-022
| 23rd February 2019
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Mahdi Cheraghchi, Valentine Kabanets, Zhenjian Lu, Dimitrios Myrisiotis#### Circuit Lower Bounds for MCSP from Local Pseudorandom Generators

Revisions: 1

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TR16-125
| 31st July 2016
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Karthekeyan Chandrasekaran, Mahdi Cheraghchi, Venkata Gandikota, Elena Grigorescu#### Local Testing for Membership in Lattices

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TR15-076
| 28th April 2015
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Mahdi Cheraghchi, Piotr Indyk#### Nearly Optimal Deterministic Algorithm for Sparse Walsh-Hadamard Transform

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TR15-030
| 6th March 2015
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Mahdi Cheraghchi, Elena Grigorescu, Brendan Juba, Karl Wimmer, Ning Xie#### ${\mathrm{AC}^{0} \circ \mathrm{MOD}_2}$ lower bounds for the Boolean Inner Product

Revisions: 1

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TR13-121
| 4th September 2013
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Mahdi Cheraghchi, Venkatesan Guruswami#### Non-Malleable Coding Against Bit-wise and Split-State Tampering

Revisions: 1

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TR13-118
| 2nd September 2013
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Mahdi Cheraghchi, Venkatesan Guruswami#### Capacity of Non-Malleable Codes

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TR12-172
| 8th December 2012
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Mahdi Cheraghchi, Anna Gal, Andrew Mills#### Correctness and Corruption of Locally Decodable Codes

Revisions: 1

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TR12-082
| 28th June 2012
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Mahdi Cheraghchi, Venkatesan Guruswami, Ameya Velingker#### Restricted Isometry of Fourier Matrices and List Decodability of Random Linear Codes

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TR11-090
| 2nd June 2011
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Mahdi Cheraghchi, Adam Klivans, Pravesh Kothari, Homin Lee#### Submodular Functions Are Noise Stable

Revisions: 2

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TR10-132
| 18th August 2010
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Mahdi Cheraghchi, Johan HÃ¥stad, Marcus Isaksson, Ola Svensson#### Approximating Linear Threshold Predicates

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TR05-070
| 6th July 2005
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Mahdi Cheraghchi#### On Matrix Rigidity and the Complexity of Linear Forms

Huck Bennett, Mahdi Cheraghchi, Venkatesan Guruswami, Joao Ribeiro

We prove that the Minimum Distance Problem (MDP) on linear codes over any fixed finite field and parameterized by the input distance bound is W[1]-hard to approximate within any constant factor. We also prove analogous results for the parameterized Shortest Vector Problem (SVP) on integer lattices. Specifically, we prove that ... more >>>

Eric Allender, Mahdi Cheraghchi, Dimitrios Myrisiotis, Harsha Tirumala, Ilya Volkovich

One-way functions (OWFs) are central objects of study in cryptography and computational complexity theory. In a seminal work, Liu and Pass (FOCS 2020) proved that the average-case hardness of computing time-bounded Kolmogorov complexity is equivalent to the existence of OWFs. It remained an open problem to establish such an equivalence ... more >>>

Mahdi Cheraghchi, Shuichi Hirahara, Dimitrios Myrisiotis, Yuichi Yoshida

For a size parameter $s\colon\mathbb{N}\to\mathbb{N}$, the Minimum Circuit Size Problem (denoted by ${\rm MCSP}[s(n)]$) is the problem of deciding whether the minimum circuit size of a given function $f \colon \{0,1\}^n \to \{0,1\}$ (represented by a string of length $N := 2^n$) is at most a threshold $s(n)$. A ... more >>>

Mahdi Cheraghchi, Valentine Kabanets, Zhenjian Lu, Dimitrios Myrisiotis

The Minimum Circuit Size Problem (MCSP) asks if a given truth table of a Boolean function $f$ can be computed by a Boolean circuit of size at most $\theta$, for a given parameter $\theta$. We improve several circuit lower bounds for MCSP, using pseudorandom generators (PRGs) that are local; a ... more >>>

Karthekeyan Chandrasekaran, Mahdi Cheraghchi, Venkata Gandikota, Elena Grigorescu

Motivated by the structural analogies between point lattices and linear error-correcting codes, and by the mature theory on locally testable codes, we initiate a systematic study of local testing for membership in lattices. Testing membership in lattices is also motivated in practice, by applications to integer programming, error detection in ... more >>>

Mahdi Cheraghchi, Piotr Indyk

For every fixed constant $\alpha > 0$, we design an algorithm for computing the $k$-sparse Walsh-Hadamard transform of an $N$-dimensional vector $x \in \mathbb{R}^N$ in time $k^{1+\alpha} (\log N)^{O(1)}$. Specifically, the algorithm is given query access to $x$ and computes a $k$-sparse $\tilde{x} \in \mathbb{R}^N$ satisfying $\|\tilde{x} - \hat{x}\|_1 \leq ... more >>>

Mahdi Cheraghchi, Elena Grigorescu, Brendan Juba, Karl Wimmer, Ning Xie

$\mathrm{AC}^{0} \circ \mathrm{MOD}_2$ circuits are $\mathrm{AC}^{0}$ circuits augmented with a layer of parity gates just above the input layer. We study the $\mathrm{AC}^{0} \circ \mathrm{MOD}_2$ circuit lower bound for computing the Boolean Inner Product functions. Recent works by Servedio and Viola (ECCC TR12-144) and Akavia et al. (ITCS 2014) have ... more >>>

Mahdi Cheraghchi, Venkatesan Guruswami

Non-malleable coding, introduced by Dziembowski, Pietrzak and Wichs (ICS 2010), aims for protecting the integrity of information against tampering attacks in situations where error-detection is impossible. Intuitively, information encoded by a non-malleable code either decodes to the original message or, in presence of any tampering, to an unrelated message. Non-malleable ... more >>>

Mahdi Cheraghchi, Venkatesan Guruswami

Non-malleable codes, introduced by Dziembowski, Pietrzak and Wichs (ICS 2010), encode messages $s$ in a manner so that tampering the codeword causes the decoder to either output $s$ or a message that is independent of $s$. While this is an impossible goal to achieve against unrestricted tampering functions, rather surprisingly ... more >>>

Mahdi Cheraghchi, Anna Gal, Andrew Mills

Locally decodable codes (LDCs) are error correcting codes with the extra property that it is sufficient to read just a small number of positions of a possibly corrupted codeword in order to recover any one position of the input. To achieve this, it is necessary to use randomness in the ... more >>>

Mahdi Cheraghchi, Venkatesan Guruswami, Ameya Velingker

We prove that a random linear code over $\mathbb{F}_q$, with probability arbitrarily close to $1$, is list decodable at radius $1-1/q-\epsilon$ with list size $L=O(1/\epsilon^2)$ and rate $R=\Omega_q(\epsilon^2/(\log^3(1/\epsilon)))$. Up to the polylogarithmic factor in $1/\epsilon$ and constant factors depending on $q$, this matches the lower bound $L=\Omega_q(1/\epsilon^2)$ for the list ... more >>>

Mahdi Cheraghchi, Adam Klivans, Pravesh Kothari, Homin Lee

We show that all non-negative submodular functions have high noise-stability. As a consequence, we obtain a polynomial-time learning algorithm for this class with respect to any product distribution on $\{-1,1\}^n$ (for any constant accuracy parameter $\epsilon$ ). Our algorithm also succeeds in the agnostic setting. Previous work on learning submodular ... more >>>

Mahdi Cheraghchi, Johan HÃ¥stad, Marcus Isaksson, Ola Svensson

We study constraint satisfaction problems on the domain $\{-1,1\}$, where the given constraints are homogeneous linear threshold predicates. That is, predicates of the form $\mathrm{sgn}(w_1 x_1 + \cdots + w_n x_n)$ for some positive integer weights $w_1, \dots, w_n$. Despite their simplicity, current techniques fall short of providing a classification ... more >>>

Mahdi Cheraghchi

The rigidity function of a matrix is defined as the minimum number of its entries that need to be changed in order to reduce the rank of the matrix to below a given parameter. Proving a strong enough lower bound on the rigidity of a matrix implies a nontrivial lower ... more >>>