Questions tagged [pseudorandomness]

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12
votes
1answer
389 views

Deterministic error reduction, state-of-the-art?

Assume one has a randomized (BPP) algorithm $A$ using $r$ bits of randomness. Natural ways to amplify its probability of success to $1-\delta$, for any chosen $\delta>0$, are Independent runs + ...
3
votes
1answer
147 views

Algebraic construction of $\varepsilon$-biased sets

Let $\ell> 1$ be an integer and consider the mapping $\text{Tr}:\mathbb{F}_{2^\ell}\to\mathbb{F}_{2^\ell}$ defined by $$\text{Tr}(x)=x^{2^0}+x^{2^{1}}+\cdots+x^{2^{\ell-1}}$$ It is then possible to ...
4
votes
1answer
111 views

Strong seeded randomness extractors with low entropy loss

I would like to implement a strong seeded randomness extractor for flat sources as a part of my project. Most of the literature on seeded extractors is concentrated on minimizing seed length. ...
1
vote
0answers
69 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 ...
3
votes
0answers
55 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\}^...
6
votes
1answer
118 views

Average-case analogue of Small-bias Spaces

Recall that an $\epsilon$-biased space is a set $S \subset \{0,1\}^n$ such that for every non-zero linear test $\alpha \in \{0,1\}^n \setminus \{0\}^n$, the expected bias $$| \mathbb{E}_{x \in S} [ (-...
2
votes
0answers
75 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 ...
1
vote
0answers
80 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, ...
4
votes
0answers
139 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
votes
0answers
91 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
votes
0answers
102 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 ...
10
votes
1answer
391 views

Can true randomness (provably) be replaced with Kolmogorov randomness for RP?

Have there been any attempts to show that Kolmogorov randomness would be sufficient for RP? Would the probability used in the statement "If the correct answer is YES, then it (the probabilistic Turing ...
10
votes
2answers
763 views

Is it known whether $BPP\cap NP\subseteq RP$?

The reverse inclusion is obvious, as is the fact that any self-reducible NP language in BPP is also in RP. Is this also known to hold for non-self-reducible NP languages?
1
vote
0answers
144 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
vote
0answers
103 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, ...
0
votes
1answer
52 views

Upper bound on the pseudoentropy of any distribution

From here: The notion of pseudoentropy is only useful, however, as a lower bound on the computational entropy in a distribution. Indeed, it can be shown that every distribution on $\{0,1\}^n$ is ...
1
vote
0answers
171 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| \...
2
votes
1answer
112 views

On the definition of pseudoentropy

$\mathbf{Definition.}$ A random variable $X$ has pseudoentropy $k$ if it is computationally indistinguishable from a random variable $Y$ with $H(Y) = k$, where $H(Y)$ is the Shannon entropy of $Y$. ...
5
votes
1answer
185 views

Simple candidates for pseudorandom permutations?

Even though it is not known whether one-way functions exist, there are several candidate functions used in practice for cryptographic applications that are efficiently computable but are conjectured ...
4
votes
1answer
230 views

Special properties of bipartite expanders

It is well known that expanders, and often the special case of bipartite expanders, have found many uses in derandomization, coding, etc. However, I am curious if there are any special properties of ...
2
votes
0answers
104 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 $D$ ...
0
votes
3answers
208 views

How are random numbers structure-less?

I'm using random numbers for simulations. The main reason is to have an input sequence where no (simulation) algorithm is going to lock on a pattern and introduce unwanted effects into the simulation. ...
12
votes
2answers
282 views

Pseudorandom generator for finite automata

Let $d$ be a constant. How can we provably construct a pseudorandom generator that fools $d$-state finite automata? Here, a $d$-state finite automata has $d$ nodes, a start node, a set of nodes ...
6
votes
1answer
221 views

Pseudorandom generators indistinguishable by uniform deterministic adversaries

I've seen pseudorandom generators defined for nonuniform efficient adversaries, or uniform probabilistic efficient adversaries. (For example, a monograph Pseudorandomness by Vadhan (here's its draft ...
20
votes
0answers
401 views

$RL=L$ 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 ...
2
votes
1answer
218 views

n irrational number whose digits are pseudo-random: conceptual mismatch?

Are there irrational numbers whose digits are considered pseudo-random? Both concepts seem to be mismatched as a pseudo-random number generator typically is periodic and therefore generates a ...
4
votes
1answer
91 views

Is being fooled by limited independence preserved by products?

Question. Let $f,g : \{\pm 1\}^n \to \{\pm 1\}$ be $\varepsilon$-fooled by $k$-wise independence -- i.e. for any $k$-wise independent random variable $X$, $\left|\mathbb{E}[f(X)] - \mathbb{E}[f(U)]\...
5
votes
0answers
119 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 ...
7
votes
1answer
213 views

Are there any distributions with only polynomially many non-zero Fourier coefficients and a small support?

For a distribution X over $\{0,1\}^n$, we can define the Fourier coefficient of the distribution as $\hat{Y}(s)= \textbf{E}_{y\in Y}({\chi_s(y)})$. The question I have is, do there exist distributions ...
4
votes
1answer
384 views

What is full-entropy bit-strings?

I was going through the description of NIST Randomness Beacon. I would like to know the meaning of the term full-entropy bit-strings used in the third paragraph.
2
votes
0answers
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 ...
7
votes
2answers
366 views

What bound can we get using $k$-th moment inequality under 3-wise independence?

Let $X_1,\dots, X_n$ be 0-1 random variables, which are $3$-wise independent. We want to give a upper bound to $\Pr(|\Sigma_iX_i-\mu|\geq t)$. Can we get better bound than $\Theta\left(\frac{1}t\right)...
10
votes
1answer
229 views

Is deterministic pseudorandomness possibly stronger than randomness in parallel?

Let the class BPNC (the combination of $\mathsf{BPP}$ and $\mathsf{NC}$) be log depth parallel algorithms with bounded error probability and access to a random source (I'm not sure if this has a ...
16
votes
3answers
380 views

What is the motivation behind the definition of pseudorandom in Nisan/Wigderson?

I am reading the classic "Hardness vs Randomness" by Nisan and Wigderson. Let $B=\{0,1\}$, and fix a function $l\colon \mathbb{N} \to \mathbb{N}$. They define a family of functions $G = \{G_n : B^{l(...
7
votes
2answers
608 views

Rigorous proof that a random function and a random permutation cannot be distinguished in polynomial time

I have a background in number theory and I'm trying to learn how to reason rigorously about algorithms. I'm reading chapter 2 of Katz and Lindell's Introduction to Modern Cryptography. Show that no ...
17
votes
2answers
885 views

Are theoretically sound pseudorandom generators used in practice?

As far as I'm aware, most implementations of pseudorandom number generation in practice use methods such as linear shift feedback registers (LSFRs), or these "Mersenne Twister" algorithms. While they ...
9
votes
0answers
174 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)-...
4
votes
1answer
231 views

Pseudo-Random Function families whose instances have full domain

The GGM construction gives (PRF) pseudo-random function families whose instance's input's are binary strings of a single length. I've convinced myself that one could get a PRF family whose instances ...
0
votes
1answer
247 views

Non-computable=>normal?

If we have an infinite string of 0's and 1's, such that no finite Turing-machine can output it. What can we say about the string? Must it be normal, ie. must every finite sequence appear infinite ...
17
votes
0answers
293 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 ...
21
votes
3answers
746 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 ...
2
votes
0answers
472 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 interval ...
10
votes
0answers
430 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 \...
2
votes
4answers
563 views

Algorithmic distinctions between random and pseudorandom.

Given a specific pseudo random number generator (e.g. Mersenne twister) $r()$ and a true random number generator $q()$ is there an algorithm $f(x,y)$ such that: $f(r(),r()) = 1$ almost always. $f(q(),...
1
vote
3answers
674 views

Simple question about pseudorandom generator

I am stuck on the following question related to pseudorandom generator and any help would be appreciated. Let $G:\{0,1\}^k \to \{0,1\}^{k+1}$ be a pseudorandom generator. Define $G':\{0,1\}^{2k}...
6
votes
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 ...
20
votes
2answers
499 views

Explicit balanced matrix

Is it possible to build an explicit $N \times N$ $0/1$-matrix with $N^{1.5}$ ones such that every $N^{0.499} \times N^{0.499}$ submatrix contains less than $N^{0.501}$ ones? Or probably it is ...
15
votes
3answers
623 views

Surveys on pseudo-random number generator design?

I am interested in generation of pseudo-random numbers for cryptography. Besides Chapter 5 of Menezes/Oorschot/Vanstone; Chapter 8 of Stinson; and Chapter 3 of Goldreich, where else could I find more? ...
17
votes
1answer
482 views

Fooling arbitrary symmetric functions

A distribution $\mathcal{D}$ is said to $\epsilon$-fool a function $f$ if $|E_{x\in U}(f(x)) - E_{x\in \mathcal{D}}(f(x))| \leq \epsilon$. And it is said to fool a class of functions if it fools every ...