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5
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0answers
69 views

Applications of small Kakeya sets over finite fields

It was proved by Dvir that a Kakeya set in $\mathbb{F}_q^n$ has size at least $q^n/n!$, a bound which was later improved to $q^n/2^n$. For $n = 2$ and $q$ odd the exact bound is $q(q+1)/2 + (q-1)/2$ ...
0
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0answers
67 views

deterministic randomness extractor and privacy

Suppose $X$ is a message which takes values on the set $\{x_1, \dots, x_m\}$ with probability distribution $P_X$. We transmit the message $X$ over the channel $P_{Y|X}$ which outputs $Y$ taking ...
4
votes
0answers
97 views

Strong extractors with reusable seeds

I have convinced myself of the following: For every $(k,\epsilon^2\hspace{.005 in})$-strong extractor Ext, for every distribution $X$, if $\;\; k\leq$ $\:H_{\infty}$$(\hspace{.01 in}X\hspace{.015 ...
3
votes
0answers
70 views

Are there any “two-sided” strong blenders?

My question is related to explicit extractors and strong blenders. We can define explicit strong blender in a straight forward way. I want to know if there are any known explicit strong blenders ...
18
votes
3answers
540 views

Running a BPP algorithm with a half-random, half-adversarial string

Consider the following model: an n-bit string r=r1...rn is chosen uniformly at random. Next, each index i∈{1,...,n} is put into a set A with independent probability 1/2. Finally, an adversary ...
9
votes
0answers
156 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 ...
20
votes
3answers
601 views

From Extractors to Pseudorandom Generators?

Luca Trevisan showed how many constructions of pseudorandom generators can in fact be thought of as extractor constructions: http://www.cs.berkeley.edu/~luca/pubs/extractor-full.pdf Is there a ...