Related papers: A Pseudorandom Generator for Polynomial Threshold …
In this paper, we consider situations in which a given logical function is realized by a multithreshold threshold function. In such situations, constant functions can be easily obtained from multithreshold threshold functions, and…
Random graph generation is an important tool for studying large complex networks. Despite abundance of random graph models, constructing models with application-driven constraints is poorly understood. In order to advance state-of-the-art…
Testing the quality of pseudorandom number generators is an important issue. Security requirements become more and more demanding, weaknesses in this matter are simply not acceptable. There is a need for an in-depth analysis of statistical…
Sequential algorithms are popular for experimental design, enabling emulation, optimisation and inference to be efficiently performed. For most of these applications bespoke software has been developed, but the approach is general and many…
Consider a bounded-degree graph $G$ that belongs to a minor-closed family (such as planar graphs). Such a graph has a hyperfinite decomposition, wherein, for a sufficiently small $\varepsilon > 0$, one can remove $\varepsilon dn$ edges to…
We study the pseudorandomness of automatic sequences in terms of well-distribution and correlation measure of order 2. We detect non-random behavior which can be derived either from the functional equations satisfied by their generating…
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 $d^{1+\epsilon(d)} \times (\log q)^{5+\epsilon(q)}$…
We present a new secret sharing algorithm that provides the storage efficiency of an Information Dispersal Algorithm (IDA) while providing perfect secret sharing. We achieve this by mixing the input message with random bytes generated using…
Polynomial approximations to boolean functions have led to many positive results in computer science. In particular, polynomial approximations to the sign function underly algorithms for agnostically learning halfspaces, as well as…
We conjecture that bounded generalised polynomial functions cannot be generated by finite automata, except for the trivial case when they are periodic away from a finite set. Using methods from ergodic theory, we are able to partially…
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 establish generalized Gaussian bounds and local limit theorems with Gaussian-type error for the convolution powers of certain complex-valued functions on $\mathbb{Z}^d$. These global space-times estimates/error, which are sharp in…
In this paper I describe two general methods of constructing a probability density function (pdf) of a Benford random variable, and show that the pdf of every Benford random variable may be constructed by these methods. These constructions…
We study the appearance of powers of Hamilton cycles in pseudorandom graphs, using the following comparatively weak pseudorandomness notion. A graph $G$ is $(\varepsilon,p,k,\ell)$-pseudorandom if for all disjoint $X$ and $Y\subset V(G)$…
Given a Gaussian Markov random field, we consider the problem of selecting a subset of variables to observe which minimizes the total expected squared prediction error of the unobserved variables. We first show that finding an exact…
We investigate the periodic structure of the exponential pseudorandom number generator obtained from the map $x\mapsto g^x\pmod p$ that acts on the set $\{1, \ldots, p-1\}$.
Goldreich suggested candidates of one-way functions and pseudorandom generators included in $\mathsf{NC}^0$. It is known that randomly generated Goldreich's generator using $(r-1)$-wise independent predicates with $n$ input variables and…
We remark on pseudo-elliptic integrals and on exceptional function fields, namely function fields defined over an infinite base field but nonetheless containing non-trivial units. Our emphasis is on some elementary criteria that must be…
The main aim of this paper is to provide a unified approach to deriving identities for the Bernstein polynomials using a novel generating function. We derive various functional equations and differential equations using this generating…
We introduce a new type of convergence in probability theory, which we call ``mod-Gaussian convergence''. It is directly inspired by theorems and conjectures, in random matrix theory and number theory, concerning moments of values of…