Questions tagged [pseudorandomness]

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21
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$RL=L$ Progress Since 2006

Reingold, Trevisan, and Vadhan's breakthrough 2006 paper (http://dl.acm.org/citation.cfm?id=1132583) reduced the problem of showing $RL=L$ to producing pseudorandom walks on regular digraphs that are ...
17
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0answers
297 views

Problem-Dependent Derandomization

The famous result of Impagliazzo and Wigderson in '97 cemented our belief that BPP is most likely the same as P; that is, problems that can be efficiently solved with randomness can also be ...
10
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0answers
435 views

Expectation of Gowers norm

This was an assignment problem in a course on analytics combinatorics that I had taken this semester. Here is the problem: Let $\mathbf{F}$ be the set of boolean functions, $f: \{0,1\}^n \rightarrow \...
9
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0answers
175 views

Pseudorandom object yielding shrinkage in $\ell_p$ norm?

Extractors have the following property: For a random variable $X$ of min-entropy $k$ and a seed $Y$, denote the output of an $(k,\epsilon)$-extractor by $\mathrm{Ext}(X,Y)$. Then $\|\mathrm{Ext}(X,Y)-...
6
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0answers
165 views

Distributions over circuits and N-to-N vs N-to-1 circuits

This is really a two part question, and they aren't necessarily related. First, my understanding of natural proof barriers is that they are based on the idea that a suitable distribution over small ...
5
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0answers
123 views

Largest size for randomness extractor

Suppose we have a source $X$ with min-entropy $\ell$. A randomness extractor is defined as a function $f$ which satisfies the total variation $||f(X, R)-U_M||_{TV}\leq \epsilon$ where $R$ is an ...
4
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0answers
140 views

More powerful generator than Nisan-Wigderson one

Nisan-Wigderson generator can be computed in $\log^{O(1)} n$ space and fools all constant-depth circuits of size poly($n$). I mean Theorem 5 here. I want another generator, that can be computed in ...
4
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0answers
92 views

Sampling Functions Efficiently vs Pseudorandom Generators

Let $X$ be a set of $n$-bit Boolean functions of the form $f:\{0,1\}^n\rightarrow \{0,1\}$. For instance, $X$ could be the set of $n$-bit monotone Boolean functions, or the set of $n$-bit functions ...
4
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0answers
106 views

Looking for an exposition of the proof of the LMN theorem

Is there any lecture note or review paper which gives a self-contained proof of the Linial-Mansour-Nisan theorem? The exposition of that in Ryan O'Donnel's book seems to use terminology and notation ...
3
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0answers
59 views

Pseudodeterministically choosing elements from efficiently samplable distributions (or, the plausibility of a weak choice principle)

Suppose we have a poly-time samplable family of distribution. I.e., a family of distributions $D_n \subseteq \{0, 1\}^{\mathsf{poly}(n)}$ and an algorithm $S$ for which $D_n = (r \leftarrow^\$ \{0,1\}^...
2
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0answers
77 views

Scaled down and scaled up versions of Impagliazzo-Wigderson Therem

A famous theorem due to Impagliazzo and Wigderson states that if some function in $E=DTIME[2^{O(n)}]$ requires circuits of size $2^{\Omega(n)}$ then P=BPP. When can we change $P$ with some ...
2
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106 views

k-wise Independence vs. Min-entropy

A distribution $D$ on $\{0,1\}^n$ is $k$-wise independent if any $k$ of the underlying $n$ random variables are independent and each is uniformly distributed. To me this looks similar in spirit to $D$ ...
2
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114 views

What upper bound can we get under 3-wise independence? (comparable edition)

Here is the original question: What bound can we get using $k$-th moment inequality under 3-wise independence? .Yury has given a 3-wise independent example that shows the upper bound is no better than ...
2
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0answers
484 views

Expanded use of an LFSR?

We want an algorithm for the following task: We are given $n$ and $i$ and we want to check if an $n$-bit LFSR with the sub-register exactly $n/2$ and an interval of $i$ "works". We say an ...
1
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0answers
74 views

Missing proof in Salil Vadhan's monograph on pseudorandomness, Random Walks and S-T Connectivity

In Salil Vadhan's monograph on pseudorandomness, chapter 2, half of the proof of Lemma 2.51 is missing http://people.seas.harvard.edu/~salil/pseudorandomness/power.pdf . I don't state the full lemma ...
1
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0answers
104 views

Time complexity of polynomial regression with random coefficients

Suppose that I have $$\lim_{k,l,m \rightarrow \infty}(f(x,l) ~~=~~ \sum_{n=1}^{m} g(a_{n,k})h(x,n)$$ where the $a_{n,k}$ are pseudo-random real numbers with $k$ digits generated in an arbitrary way, ...
1
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0answers
168 views

How does one sample uniformly at random from an uncountably infinite set?

I want to know if there are any examples of polynomial time algorithms which can sample uniformly at random from a given uncountably infinite set? (assuming it is possible) Does it help if the sample ...
1
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0answers
105 views

Extractor with somewhat corrupted seeds

In conditional min-entropy extractor, there is a joint distribution $(X,Y)$ such that if the average min-entropy (for some appropriate notion of it) ${\rm H}_\infty(X|Y)$ is large, then ${\rm Ext}(X, ...
1
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184 views

A question about combinatorial design in Nisan-Wigderson Generator

Let $[d]$ be a universe and $S_1, \dots, S_m$ be an $(\ell, a)$-design over $[d]$ which means that: $\forall i \in [m], S_i \subseteq [d], |S_i|=\ell$. $\forall i \neq j \in [m]$, $|S_i \cap S_j| \...