Scrambled Linear Pseudorandom Number Generators

TL;DR

This paper introduces new efficient F_2-linear transformations and nonlinear scramblers to improve the statistical quality of pseudorandom number generators, achieving high speed, small memory footprint, and passing rigorous tests.

Contribution

The paper presents novel F_2-linear transformations and scramblers, with theoretical proofs, to enhance the statistical properties of pseudorandom generators.

Findings

New F_2-linear transformations with good statistical properties

Effective nonlinear scramblers reduce linear artifacts

Generators are extremely fast, small, and pass strong statistical tests

Abstract

$\mathbf F_2$-linear pseudorandom number generators are very popular due to their high speed, to the ease with which generators with a sizable state space can be created, and to their provable theoretical properties. However, they suffer from linear artifacts that show as failures in linearity-related statistical tests such as the binary-rank and the linear-complexity test. In this paper, we give two new contributions. First, we introduce two new $\mathbf F_2$-linear transformations that have been handcrafted to have good statistical properties and at the same time to be programmable very efficiently on superscalar processors, or even directly in hardware. Then, we describe some scramblers, that is, nonlinear functions applied to the state array that reduce or delete the linear artifacts, and propose combinations of linear transformations and scramblers that give extremely fast pseudorandom number generators of high quality. A novelty in our approach is that we use ideas from the theory of filtered linear-feedback shift registers to prove some properties of our scramblers, rather than relying purely on heuristics. In the end, we provide simple, extremely fast generators that use a few hundred bits of memory, have provable properties, and pass strong statistical tests.