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Electronic Colloquium on Computational Complexity

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Reports tagged with average-case hardness:
TR05-043 | 5th April 2005
Emanuele Viola

Pseudorandom Bits for Constant-Depth Circuits with Few Arbitrary Symmetric Gates

We exhibit an explicitly computable `pseudorandom' generator stretching $l$ bits into $m(l) = l^{\Omega(\log l)}$ bits that look random to constant-depth circuits of size $m(l)$ with $\log m(l)$ arbitrary symmetric gates (e.g. PARITY, MAJORITY). This improves on a generator by Luby, Velickovic and Wigderson (ISTCS '93) that achieves the same ... more >>>

TR05-154 | 11th December 2005
Albert Atserias

Non-Uniform Hardness for NP via Black-Box Adversaries

We may believe SAT does not have small Boolean circuits.
But is it possible that some language with small circuits
looks indistiguishable from SAT to every polynomial-time
bounded adversary? We rule out this possibility. More
precisely, assuming SAT does not have small circuits, we
show that ... more >>>

TR09-143 | 22nd December 2009
Noam Livne

On the Construction of One-Way Functions from Average Case Hardness

In this paper we study the possibility of proving the existence of
one-way functions based on average case hardness. It is well-known
that if there exists a polynomial-time sampler that outputs
instance-solution pairs such that the distribution on the instances
is hard on average, then one-way functions exist. We study ... more >>>

TR21-009 | 1st February 2021
Eric Allender, Mahdi Cheraghchi, Dimitrios Myrisiotis, Harsha Tirumala, Ilya Volkovich

One-way Functions and Partial MCSP

Revisions: 1 , Comments: 1

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 >>>

TR21-040 | 15th March 2021
Lijie Chen, Zhenjian Lu, Xin Lyu, Igor Carboni Oliveira

Majority vs. Approximate Linear Sum and Average-Case Complexity Below NC1

We develop a general framework that characterizes strong average-case lower bounds against circuit classes $\mathcal{C}$ contained in $\mathrm{NC}^1$, such as $\mathrm{AC}^0[\oplus]$ and $\mathrm{ACC}^0$. We apply this framework to show:

- Generic seed reduction: Pseudorandom generators (PRGs) against $\mathcal{C}$ of seed length $\leq n -1$ and error $\varepsilon(n) = n^{-\omega(1)}$ can ... more >>>

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