Related papers: Discrete logarithms in free groups

This paper considers approximate smoothing for discretely observed non-linear stochastic differential equations. The problem is tackled by developing methods for linearising stochastic differential equations with respect to an arbitrary…

This paper presents a means with time complexity of at worst O(n^3) to compute the discrete logarithm on cyclic finite groups of integers modulo p. The algorithm makes use of reduction of the problem to that of finding the concurrent zeros…

The central limit theorem ensures that a sum of random variables tends to a Gaussian distribution as their total number tends to infinity. However, for a class of positive random variables, we find that the sum tends faster to a log-normal…

We summarize our renormalization group approach for the vector model as well as the matrix model which are the discretized quantum gravity in one- and two-dimensional spacetime. A difference equation is obtained which relates free energies…

According to the well-known Heyde theorem, the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given another. In the article, we…

We study microscopic observables of the Discrete Gaussian model (i.e., the Gaussian free field restricted to take integer values) at high temperature using the renormalisation group method. In particular, we show the central limit theorem…

We address two questions of Simon Thomas. First, we show that for any n>2 one can find a four generated free subgroup of SLn(Z) which is profinitely dense. More generally, we show that an arithmetic group \Gamma which admits the congruence…

The Pollard Rho algorithm is a widely used algorithm for solving discrete logarithms on general cyclic groups, including elliptic curves. Recently the first nontrivial runtime estimates were provided for it, culminating in a sharp…

This work investigates the inverse problem of generator recovery in matrix Lie groups from discretized trajectories. Let $G$ be a real matrix Lie group and $\mathfrak{g} = \text{Lie}(G)$ its corresponding Lie algebra. A smooth trajectory…

We study limit distributions of independent random matrices as well as limit joint distributions of their blocks under normalized partial traces composed with classical expectation. In particular, we are concerned with the ensemble of…

Given a local field $\widehat K$ with positive characteristic, we study the dynamics of the diagonal subgroup of the linear group $\operatorname{GL}_n(\widehat K)$ on homogeneous spaces of discrete lattices in ${\widehat K}^{\,n}$. We first…

We show that the Riemannian Gaussian distributions on symmetric spaces, introduced in recent years, are of standard random matrix type. We exploit this to compute analytically marginals of the probability density functions. This can be done…

The discrete logarithm problem is a fundamental challenge in number theory with significant implications for cryptographic protocols. In this paper, we investigate the limitations of gradient-based methods for learning the parity bit of the…

We present discretize_distributions, a Python package that efficiently constructs discrete approximations of Gaussian mixture distributions and provides guarantees on the approximation error in Wasserstein distance. The package implements…

We prove a discretized Product Theorem for general simple Lie groups, in the spirit of Bourgain's Discretized Sum-Product Theorem.

A so-called Renormalization Group (RG) analysis is performed in order to shed some light on why the density of prime numbers in $\Bbb N^*$ decreases like the single power of the inverse neperian logarithm.

We describe all Gaussian generating functionals on several easy quantum groups given by non-crossing partitions. This includes in particular the free unitary, orthogonal and symplectic quantum groups. We further characterize central…

We consider the problem of distributedly estimating Gaussian processes in multi-agent frameworks. Each agent collects few measurements and aims to collaboratively reconstruct a common estimate based on all data. Agents are assumed with…

In Bayesian nonparametric models, Gaussian processes provide a popular prior choice for regression function estimation. Existing literature on the theoretical investigation of the resulting posterior distribution almost exclusively assume a…

In this work, we give efficient algorithms for privately estimating a Gaussian distribution in both pure and approximate differential privacy (DP) models with optimal dependence on the dimension in the sample complexity. In the pure DP…