Sampling from a discrete Gaussian distribution is an indispensable part of lattice-based cryptography. Several recent works have shown that the timing leakage from a non-constant-time implementation of the discrete Gaussian sampling algorithm could be exploited to recover the secret. In this paper, we propose a constant-time implementation of the Knuth-Yao random walk algorithm for performing c...

We analyze the convergence of the spectrum of large random graphs to the spectrum of a limit infinite graph. We apply these results to graphs converging locally to trees and derive a new formula for the Stieljes transform of the spectral measure of such graphs. We illustrate our results on the uniform regular graphs, Erdös-Renyi graphs and preferential attachment graphs. We sketch examples of a...

Quantum walks can be considered as a generalized version of the classical random walk. There are two classes of quantum walks, that is, the discrete-time (or coined) and the continuous-time quantum walks. This manuscript treats the discrete case in Part I and continuous case in Part II, respectively. Most of the contents are based on our results. Furthermore, papers on quantum walks are listed ...

We prove a shape theorem for the internal (graph) distance on the interlacement set I of the random interlacement model on Z, d ≥ 3. We provide large deviation estimates for the internal distance of distant points in this set, and use these estimates to study the internal distance on the range of a simple random walk on a discrete torus.

A model for a 1–dimensional delayed random walk is developed by generalizing the Ehrenfest model of a discrete random walk evolving on a quadratic, or harmonic, potential to the case of non–zero delay. The Fokker–Planck equation derived from this delayed random walk (DRW) is identical to that obtained starting from the delayed Langevin equation, i.e. a first–order stochastic delay differential ...

We consider a random walk on the discrete cylinder (Z/NZ)×Z, d ≥ 3 with drift N in the Z-direction and investigate the large N -behavior of the disconnection time T disc N , defined as the first time when the trajectory of the random walk disconnects the cylinder into two infinite components. We prove that, as long as the drift exponent α is strictly greater than 1, the asymptotic behavior of T...

Recently, a formalism for discrete time open quantum walks was introduced [1]. The formalism suggested is similar to the formalism of quantum Markov chains and rests upon the implementation of appropriate completely positive maps. The formalism of the open quantum random walks (OQW) includes the classical random walk and through a physical realization procedure a connection to the unitary quant...