Random Numbers from a Delay Equation

TL;DR

This paper explores using solutions of delay differential equations as a basis for random number generation, demonstrating it can pass rigorous statistical tests and comparing its performance to established methods.

Contribution

It introduces a novel RNG approach based on chaotic delay differential equations and evaluates its statistical quality and performance relative to existing generators.

Findings

The delay equation-based RNG passes TestU01's Big Crush tests.

The new generator is approximately 7 times slower than MRG32k3a.

There is no fundamental limit on the number of values this method can generate.

Abstract

Delay differential equations (DDE) can have "chaotic" solutions that can be used to mimic Brownian motion. Since a Brownian motion is random in its velocity, it is reasonable to think that a random number generator (RNG) might be constructed from such a model. In this preliminary study, we consider one specific example of this and show that it satisfies criteria commonly employed in the testing of random number generators (from TestU01's very stringent "Big Crush" battery of tests). A technique termed digit discarding, commonly used in both this generator and physical RNG's using laser feedback systems, is discussed with regard to the maximal Lyapunov exponent. Also, we benchmark the generator to a contemporary common method: the multiple recursive generator, MRG32k3a. Although our method is about 7 times slower than MRG32k3a, there is in principle no apparent limit on the number of possible values that can be generated from the scheme we present here.