Related papers: A Pseudorandom Generator for Functions of Low-Degr…

We present a randomized algorithm that on input a finite field $K$ with $q$ elements and a positive integer $d$ outputs a degree $d$ irreducible polynomial in $K[x]$. The running time is $d^{1+\epsilon(d)} \times (\log q)^{5+\epsilon(q)}$…

Rubinfeld & Vasilyan recently introduced the framework of testable learning as an extension of the classical agnostic model. It relaxes distributional assumptions which are difficult to verify by conditions that can be checked efficiently…

The problem of constructing pseudorandom generators that fool halfspaces has been studied intensively in recent times. For fooling halfspaces over the hypercube with polynomially small error, the best construction known requires seed-length…

We propose a novel pseudorandom number generator based on R\"ossler attractor and bent Boolean function. We estimated the output bits properties by number of statistical tests. The results of the cryptanalysis show that the new pseudorandom…

We initiate the study of active learning polynomial threshold functions (PTFs). While traditional lower bounds imply that even univariate quadratics cannot be non-trivially actively learned, we show that allowing the learner basic access to…

This work considers the deployment of pseudo-random number generators (PRNGs) on graphics processing units (GPUs), developing an approach based on the xorgens generator to rapidly produce pseudo-random numbers of high statistical quality.…

For $S \subseteq \{0,1\}^n$ a Boolean function $f \colon S \to \{-1,1\}$ is a polynomial threshold function (PTF) of degree $d$ and weight $W$ if there is a polynomial $p$ with integer coefficients of degree $d$ and with sum of absolute…

We provide asymptotically sharp bounds for the Gaussian surface area and the Gaussian noise sensitivity of polynomial threshold functions. In particular we show that if $f$ is a degree-$d$ polynomial threshold function, then its Gaussian…

We consider generating functionals for computing correlators in quantum field theories with random potentials. Examples of such theories include condensed matter systems with quenched disorder (e.g. spin glass) or cosmological systems in…

In this paper, we focus on analyzing the period distribution of the inversive pseudorandom number generators (IPRNGs) over finite field $({\rm Z}_{N},+,\times)$, where $N>3$ is a prime. The sequences generated by the IPRNGs are transformed…

We obtain new explicit pseudorandom generators for several computational models involving groups. Our main results are as follows: 1. We consider read-once group-products over a finite group $G$, i.e., tests of the form $\prod_{i=1}^n…

A generator of spatio-temporal pseudo-random Gaussian fields that satisfy the "proportionality of scales" property (Tsyroulnikov, 2001) is presented. The generator is based on a third-order in time stochastic differential equation with a…

Generative Adversarial Networks (GANs) have become the gold standard when it comes to learning generative models for high-dimensional distributions. Since their advent, numerous variations of GANs have been introduced in the literature,…

The probabilistic degree of a Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is defined to be the smallest $d$ such that there is a random polynomial $\mathbf{P}$ of degree at most $d$ that agrees with $f$ at each point with high…

We consider pseudo-involutions in the Riordan group where the generating function $g$ for the first column of a Riordan array satisfies a palindromic or near-palindromic functional equation. For those types of equations, we find, for very…

We study whether a uniformly random Boolean function $f : \{-1,1\}^p \to \{-1,1\}$ is determined by its Walsh--Fourier coefficients of degree at most $d$. We show that the threshold lies at $p/2$ up to an $O(\sqrt{p \log p})$ window: if \[…

We analyze the structure of the periodic trajectories of the matrix generator of pseudorandom numbers which has been proposed earlier. The structure of the periodic trajectories becomes more transparent when the rational sublattice…

We make progress on some questions related to polynomial approximations of ${\rm AC}^0$. It is known, by works of Tarui (Theoret. Comput. Sci. 1993) and Beigel, Reingold, and Spielman (Proc. $6$th CCC, 1991), that any ${\rm AC}^0$ circuit…

Consider a subfield of the field of rational functions in several indeterminates. We present an algorithm that, given a set of generators of such a subfield, finds a simple generating set. We provide an implementation of the algorithm and…

The pseudo-random number generators (PRNGs), sampling algorithms, and algorithms for generating random integers in some common statistical packages and programming languages are unnecessarily inaccurate, by an amount that may matter for…