Pseudorandom Vector Generation Using Elliptic Curves And Applications

TL;DR

This paper introduces an elliptic curve-based algorithm for generating high-dimensional pseudorandom vectors uniformly distributed in [0,1]^d, with applications in simulating Wiener processes for solving complex PDEs.

Contribution

It presents a novel, efficient method for pseudorandom vector generation using elliptic curves, enabling improved simulation of stochastic processes.

Findings

Efficient high-dimensional pseudorandom vector generation.

Application to simulating Wiener process sample paths.

Potential for numerical solutions of semilinear PDEs.

Abstract

In this paper we present, using the arithmetic of elliptic curves over finite fields, an algorithm for the efficient generation of a sequence of uniform pseudorandom vectors in high dimensions, that simulates a sample of a sequence of i.i.d. random variables, with values in the hypercube $[0,1]^d$ with uniform distribution. As an application, we obtain, in the discrete time simulation, an efficient algorithm to simulate, uniformly distributed sample path sequence of a sequence of independent standard Wiener processes. This could be employed for use, in the full history recursive multi-level Picard approximation method, for numerically solving the class of semilinear parabolic partial differential equations of the Kolmogorov type.