相关论文: A deterministic version of Pollard's p-1 algorithm

Although a deterministic polytime algorithm for primality testing is now known, the Rabin-Miller randomized test of primality continues being the most efficient and widely used algorithm. We prove the correctness of the Rabin-Miller…

Based on the observation that $\mathbb{Q}^{(p-1) \times (p-1)}$ is isomorphic to a quotient skew polynomial ring, we propose a new method for $(p-1)\times (p-1)$ matrix multiplication over $\mathbb{Q}$, where $p$ is a prime number. The main…

We present a quantum version of the classical probabilistic algorithms $\grave{a}$ la Rabin. The quantum algorithm is based on the essential use of Grover's operator for the quantum search of a database and of Shor's Fourier transform for…

We give reconstruction algorithms for subclasses of depth-3 arithmetic circuits. In particular, we obtain the first efficient algorithm for finding tensor rank, and an optimal tensor decomposition as a sum of rank-one tensors, when given…

In this paper, we combine the operator splitting methodology for abstract evolution equations with that of stochastic methods for large-scale optimization problems. The combination results in a randomized splitting scheme, which in a given…

We present an algorithm for isolating the roots of an arbitrary complex polynomial $p$ that also works for polynomials with multiple roots provided that the number $k$ of distinct roots is given as part of the input. It outputs $k$ pairwise…

In this paper, a randomized algorithm for deciding the irreducibility of an irreducible polynomial and factoring a reducible polynomial over the field of rational numbers is presented. The main idea underlying the algorithm is based on…

We study the problem of differentially private continual counting in the unbounded setting where the input size $n$ is not known in advance. Current state-of-the-art algorithms based on optimal instantiations of the matrix mechanism cannot…

Let $a,b\in \mathbb{N}$ be fixed and coprime such that $a>b$, and let $N$ be any number of the form $a^n\pm b^n$, $n\in\mathbb{N}$. We will generalize a result of Bostan, Gaudry and Schost and prove that we may compute the prime…

The number of steps any classical computer requires in order to find the prime factors of an $l$-digit integer $N$ increases exponentially with $l$, at least using algorithms known at present. Factoring large integers is therefore…

This note introduces a new class of integer factoring algorithms. Two versions of this method will be described, deterministic and probabilistic. These algorithms are practical, and can factor large classes of balanced integers N = pq, p <…

We present an algorithm which allows to solve analytically linear systems of differential equations which factorize to first order. The solution is given in terms of iterated integrals over an alphabet where its structure is implied by the…

This paper presents the concept of digit polynomials, which leads to a deterministic and unconditional integer factorization algorithm with the runtime complexity $\mathcal{O}(N^{1/4+\epsilon})$. Strassen's well known factoring approach is…

We study the efficient learnability of low-degree polynomial threshold functions (PTFs) in the presence of a constant fraction of adversarial corruptions. Our main algorithmic result is a polynomial-time PAC learning algorithm for this…

We give efficient deterministic algorithms for converting randomized query algorithms into deterministic ones. We first give an algorithm that takes as input a randomized $q$-query algorithm $R$ with description length $N$ and a parameter…

Frank-Wolfe methods are popular for optimization over a polytope. One of the reasons is because they do not need projection onto the polytope but only linear optimization over it. To understand its complexity, Lacoste-Julien and Jaggi…

Let $f(x)\in \mathbb{F}_q[x]$ be an irreducible polynomial of degree $m$ and exponent $e$, and $n$ be a positive integer such that $\nu_p(q-1)\ge \nu_{p}(e)+\nu_p(n)$ for all $p$ prime divisor of $n$. We show a fast algorithm to determine…

Many applications in scientific computing and data science require the computation of a rank-revealing factorization of a large matrix. In many of these instances the classical algorithms for computing the singular value decomposition are…

Let $\mathrm{R}$ be a real closed field and $\mathrm{D} \subset \mathrm{R}$ an ordered domain. We consider the algorithmic problem of computing the generalized Euler-Poincar\'e characteristic of real algebraic as well as semi-algebraic…

We give a deterministic algorithm that, given a composite number $N$ and a target order $D \ge N^{1/6}$, runs in time $D^{1/2+o(1)}$ and finds either an element $a \in \mathbb{Z}_N^*$ of multiplicative order at least $D$, or a nontrivial…