Michael Alekhnovich, Eli Ben-Sasson, Alexander Razborov, Avi Wigderson

We call a pseudorandom generator $G_n:\{0,1\}^n\to \{0,1\}^m$ {\em

hard} for a propositional proof system $P$ if $P$ can not efficiently

prove the (properly encoded) statement $G_n(x_1,\ldots,x_n)\neq b$ for

{\em any} string $b\in\{0,1\}^m$. We consider a variety of

``combinatorial'' pseudorandom generators inspired by the

Nisan-Wigderson generator on the one hand, and ...
more >>>

Vered Rosen

Assuming the inractability of factoring, we show that the

output of the exponentiation modulo a composite function

$f_{N,g}(x)=g^x\bmod N$ (where $N=P\cdot Q$) is pseudorandom,

even when its input is restricted to be half the size.

This result is equivalent to the simultaneous hardness of

the ...
more >>>

Emanuele Viola

We study the complexity of building

pseudorandom generators (PRGs) from hard functions.

We show that, starting from a function f : {0,1}^l -> {0,1} that

is mildly hard on average, i.e. every circuit of size 2^Omega(l)

fails to compute f on at least a 1/poly(l)

fraction of inputs, we can ...
more >>>

Emanuele Viola

We study pseudorandom generator (PRG) constructions $G^f : {0,1}^l \to {0,1}^{l+s}$ from one-way functions $f : {0,1}^n \to {0,1}^m$. We consider PRG constructions of the form $G^f(x) = C(f(q_{1}) \ldots f(q_{poly(n)}))$

where $C$ is a polynomial-size constant depth circuit

and $C$ and the $q$'s are generated from $x$ arbitrarily.

more >>>

Emanuele Viola

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 >>>

Andrej Bogdanov, Emanuele Viola

We present a new approach to constructing pseudorandom generators that fool low-degree polynomials over finite fields, based on the Gowers norm. Using this approach, we obtain the following main constructions of explicitly computable generators $G : \F^s \to \F^n$ that fool polynomials over a prime field $\F$:

\begin{enumerate}

\item a ...
more >>>

Emanuele Viola

We prove that the sum of $d$ small-bias generators $L

: \F^s \to \F^n$ fools degree-$d$ polynomials in $n$

variables over a prime field $\F$, for any fixed

degree $d$ and field $\F$, including $\F = \F_2 =

{0,1}$.

Our result improves on both the work by Bogdanov and

Viola ...
more >>>

Ilias Diakonikolas, Parikshit Gopalan, Ragesh Jaiswal, Rocco Servedio, Emanuele Viola

We show that any distribution on {-1,1}^n that is k-wise independent fools any halfspace h with error \eps for k = O(\log^2(1/\eps)/\eps^2). Up to logarithmic factors, our result matches a lower bound by Benjamini, Gurel-Gurevich, and Peled (2007) showing that k = \Omega(1/(\eps^2 \cdot \log(1/\eps))). Using standard constructions of k-wise ... more >>>

Venkatesan Guruswami, Adam Smith

In this paper, we consider coding schemes for computationally bounded channels, which can introduce an arbitrary set of errors as long as (a) the fraction of errors is bounded with high probability by a parameter p and (b) the process which adds the errors can be described by a sufficiently ... more >>>

Emanuele Viola

We show that the promise problem of distinguishing $n$-bit strings of hamming weight $\ge 1/2 + \Omega(1/\log^{d-1} n)$ from strings of weight $\le 1/2 - \Omega(1/\log^{d-1} n)$ can be solved by explicit, randomized (unbounded-fan-in) poly(n)-size depth-$d$ circuits with error $\le 1/3$, but cannot be solved by deterministic poly(n)-size depth-$(d+1)$ circuits, ... more >>>

Bill Fefferman, Ronen Shaltiel, Chris Umans, Emanuele Viola

The {\em hybrid argument}

allows one to relate

the {\em distinguishability} of a distribution (from

uniform) to the {\em

predictability} of individual bits given a prefix. The

argument incurs a loss of a factor $k$ equal to the

bit-length of the

distributions: $\epsilon$-distinguishability implies only

$\epsilon/k$-predictability. ...
more >>>

Periklis Papakonstantinou, Guang Yang

Every pseudorandom generator is in particular a one-way function. If we only consider part of the output of the

pseudorandom generator is this still one-way? Here is a general setting formalizing this question. Suppose

$G:\{0,1\}^n\rightarrow \{0,1\}^{\ell(n)}$ is a pseudorandom generator with stretch $\ell(n)> n$. Let $M_R\in\{0,1\}^{m(n)\times \ell(n)}$ be a linear ...
more >>>

Anat Ganor, Ran Raz

In 1989, Babai, Nisan and Szegedy [BNS92] gave a construction of a pseudorandom generator for logspace, based on lower bounds for multiparty communication complexity. The seed length of their pseudorandom generator was $2^{\Theta(\sqrt n)}\,\,\,$, because the best lower bounds for multiparty communication complexity are relatively weak. Subsequently, pseudorandom generators for ... more >>>

Pooya Hatami, Avishay Tal

A Boolean function is said to have maximal sensitivity $s$ if $s$ is the largest number of Hamming neighbors of a point which differ from it in function value. We construct a pseudorandom generator with seed-length $2^{O(\sqrt{s})} \cdot \log(n)$ that fools Boolean functions on $n$ variables with maximal sensitivity at ... more >>>

Rohit Gurjar, Ben Lee Volk

We construct a pseudorandom generator which fools read-$k$ oblivious branching programs and, more generally, any linear length oblivious branching program, assuming that the sequence according to which the bits are read is known in advance. For polynomial width branching programs, the seed lengths in our constructions are $\tilde{O}(n^{1-1/2^{k-1}})$ (for the ... more >>>

Eshan Chattopadhyay, Anindya De, Rocco Servedio

We show that a very simple pseudorandom generator fools intersections of $k$ linear threshold functions (LTFs) and arbitrary functions of $k$ LTFs over $n$-dimensional Gaussian space.

The two analyses of our PRG (for intersections versus arbitrary functions of LTFs) are quite different from each other and from previous analyses of ... more >>>

Fu Li, David Zuckerman

We study the task of seedless randomness extraction from recognizable sources, which are uniform distributions over sets of the form {x : f(x) = v} for functions f in some specified class C. We give two simple methods for constructing seedless extractors for C-recognizable sources.

Our first method shows that ...
more >>>

Eshan Chattopadhyay, Pooya Hatami, Shachar Lovett, Avishay Tal

A recent work of Chattopadhyay et al. (CCC 2018) introduced a new framework for the design of pseudorandom generators for Boolean functions. It works under the assumption that the Fourier tails of the Boolean functions are uniformly bounded for all levels by an exponential function. In this work, we design ... more >>>