On the $\mathbb{F}_2$-linear relations of Mersenne Twister pseudorandom number generators

TL;DR

This paper investigates the linear relations in Mersenne Twister generators using a new metric, revealing low-weight relations in MT19937 and improvements in variants like WELL generators.

Contribution

It introduces a new figure of merit, $N_v$, to assess high-dimensional non-random patterns in $ ext{F}_2$-linear generators and analyzes their implications.

Findings

MT19937 exhibits low-weight $ ext{F}_2$-linear relations beyond dimension 623.

Some output vectors are rejected or fail birthday spacings tests.

Variants like WELL generators show significant improvements in $N_v$.

Abstract

Sequence generators obtained by linear recursions over the two-element field $\mathbb{F}_2$, i.e., $\mathbb{F}_2$-linear generators, are widely used as pseudorandom number generators. For example, the Mersenne Twister MT19937 is one of the most successful applications. An advantage of such generators is that we can assess them quickly by using theoretical criteria, such as the dimension of equidistribution with $v$-bit accuracy. To compute these dimensions, several polynomial-time lattice reduction algorithms have been proposed in the case of $\mathbb{F}_2$-linear generators. In this paper, in order to assess non-random bit patterns in dimensions that are higher than the dimension of equidistribution with $v$-bit accuracy,we focus on the relationship between points in the Couture--L'Ecuyer dual lattices and $\mathbb{F}_2$-linear relations on the most significant $v$ bits of output sequences, and consider a new figure of merit $N_v$ based on the minimum weight of $\mathbb{F}_2$-linear relations whose degrees are minimal for $v$. Next, we numerically show that MT19937 has low-weight $\mathbb{F}_2$-linear relations in dimensions higher than 623, and show that some output vectors with specific lags are rejected or have small $p$-values in the birthday spacings tests. We also report that some variants of Mersenne Twister, such as WELL generators, are significantly improved from the perspective of $N_v$.