Using numerical simulation, we measured the performance of several potential quantum algorithms, based on quantum random walks, to solve Boolean satisfiability (SAT) problems. We develop the fundamentals of quantum computing and the theory of classical algorithms to indicate how these algorithms could be implemented. We also discuss the development of quantum random walks and the principles tha...

Elliptic Curve Cryptography provides similar strength of protection comparing other public key cryptosystems but requires significantly smaller key size. This paper proposes a new faster scalar multiplication algorithm aiming at a more secured Elliptic Curve Cryptography scheme. This paper also proposes a novel Elliptic Curve Cryptography scheme where maximum length random sequence generation m...

2 Binary search trees 2 2.1 Definition of a binary search tree . . . . . . . . . . . . . . . . . . 2 2.2 Profile. Discrete martingale . . . . . . . . . . . . . . . . . . . . . 3 2.3 Embedding in continuous time. Yule tree . . . . . . . . . . . . . . 4 2.4 Martingale connection Yule tree binary search tree . . . . . . . . 5 2.5 Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

We investigate the distribution of the time spent by a random walker to the right of a boundary moving with constant velocity v. For the continuous-time problem (Brownian motion), we provide a simple alternative proof of Newman's recent result [J. Phys. A 34, L89 (2001)] using a method due to Kac. We then discuss the same problem for the case of a random walk in discrete time with an arbitrary ...

In this paper, we will explore the properties of the Heat Equation on Discrete Networks, laying out groundwork and giving general forms for solutions, and then exploring the inverse problem. We will also focus on comparisons of Heat Networks with Electrical and Random-Walk Networks.

We study the asymptotic behavior for large N of the disconnection time TN of simple random walk on the discrete cylinder (Z/NZ)d × Z. When d is sufficiently large, we are able to substantially improve the lower bounds on TN previously derived in [3], for d ≥ 2. We show here that the laws of N/TN are tight.

Abstract We revisit the statistics of extremes and records symmetric random walks with stochastic resetting, extending earlier studies in several directions. put forward a diffusive scaling regime (symmetric step length distribution finite variance, weak resetting probability) where maximum walk number its up to discrete time n become asymptotically proportional each other for single typical tr...

We first obtain by analogy with the continuous (differential) case the general solution of the discrete Riccati equation. Moreover, we establish the full equivalence between our discrete Riccati equation and a corresponding homogeneous second order discrete linear equation. Our results can be considered the discrete analog of Mielnik's construction in supersymmetric quantum mechanics [J. Math. ...