Related papers: Optimal PRGs for Low-Degree Polynomials over Polyn…
We obtain tight bounds for the minimal number of generators of an ideal with bounded-degree generators in a polynomial ring $K[X_1,\dots,X_n],$ as well as a sharp quantification of the maximum possible size of a minimal generating set of…
We give a pseudorandom generator that fools degree-$d$ polynomial threshold functions over $n$-dimensional Gaussian space with seed length $\mathrm{poly}(d)\cdot \log n$. All previous generators had a seed length with at least a $2^d$…
We construct pseudorandom generators of seed length $\tilde{O}(\log(n)\cdot \log(1/\epsilon))$ that $\epsilon$-fool ordered read-once branching programs (ROBPs) of width $3$ and length $n$. For unordered ROBPs, we construct pseudorandom…
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 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.…
A degree-$d$ polynomial $p$ in $n$ variables over a field $\F$ is {\em equidistributed} if it takes on each of its $|\F|$ values close to equally often, and {\em biased} otherwise. We say that $p$ has a {\em low rank} if it can be expressed…
We present a new approach to constructing unconditional pseudorandom generators against classes of functions that involve computing a linear function of the inputs. We give an explicit construction of a pseudorandom generator that fools the…
We show that in a polynomial ring $R$ in $N$ variables over an algebraically closed field $K$ of arbitrary characteristic, any $K$-subalgebra of $R$ generated over $K$ by at most $n$ forms of degree at most $d$ is contained in a…
We study correlation bounds and pseudorandom generators for depth-two circuits that consist of a $\mathsf{SYM}$-gate (computing an arbitrary symmetric function) or $\mathsf{THR}$-gate (computing an arbitrary linear threshold function) that…
The problem of testing low-degree polynomials has received significant attention over the years due to its importance in theoretical computer science, and in particular in complexity theory. The problem is specified by three parameters:…
Introduced in [CG24], pseudorandom error-correcting codes (PRCs) are a new cryptographic primitive with applications in watermarking generative AI models. These are codes where a collection of polynomially many codewords is computationally…
We study weighted pseudorandom generators (WPRGs) and derandomizations for read-once branching programs (ROBPs). Denote $n$ and $w$ as the length and the width of a ROBP. We have the following results. For standard ROBPs, we give an…
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…
Low-degree polynomials have emerged as a powerful paradigm for providing evidence of statistical-computational gaps across a variety of high-dimensional statistical models [Wein25]. For detection problems -- where the goal is to test a…
High quality random numbers are necessary in the modern world. Ranging from encryption keys in cyber security to models and simulations for scientific use: it's important that these random numbers are of high quality and quickly attainable.…
A technique for controlling errors in the functioning of nodes for the formation of $q$-valued pseudo-random sequences (PRS) operating under both random errors and errors generated through intentional attack by an attacker is provided, in…
A method for generating irreducible polynomials of degree n over the finite field GF(2) is proposed. The irreducible polynomials are found by solving a system of equations that brings the information on the internal properties of the…
We study the problem of learning a low-degree spherical polynomial of degree $k_0 = \Theta(1) \ge 1$ defined on the unit sphere in $\RR^d$ by training an over-parameterized two-layer neural network with augmented feature in this paper. Our…
Suppose $I$ is an ideal of a polynomial ring over a field, $I\subseteq k[x_1,\ldots,x_n]$, and whenever $fg\in I$ with degree $\leq b$, then either $f\in I$ or $g\in I$. When $b$ is sufficiently large, it follows that $I$ is prime.…
We consider the construction of maximal families of polynomials over the finite field $\mathbb{F}_q$, all having the same degree $n$ and a nonzero constant term, where the degree of the GCD of any two polynomials is $d$ with $1 \le d\le n$.…