Linear Transformations for Randomness Extraction

TL;DR

This paper investigates the use of linear transformations, especially sparse random matrices, for efficient randomness extraction from various sources, emphasizing their optimality, flexibility, and hardware suitability for high-speed applications.

Contribution

It demonstrates the asymptotic optimality of sparse random matrices for extracting randomness and explores explicit matrix constructions like BCH codes, balancing efficiency and computational complexity.

Findings

Sparse random matrices are asymptotically optimal for independent and bit-fixing sources.

Linear transformations preserve information effectively across different source models.

Explicit BCH code matrices offer good extraction but are computationally intensive.

Abstract

Information-efficient approaches for extracting randomness from imperfect sources have been extensively studied, but simpler and faster ones are required in the high-speed applications of random number generation. In this paper, we focus on linear constructions, namely, applying linear transformation for randomness extraction. We show that linear transformations based on sparse random matrices are asymptotically optimal to extract randomness from independent sources and bit-fixing sources, and they are efficient (may not be optimal) to extract randomness from hidden Markov sources. Further study demonstrates the flexibility of such constructions on source models as well as their excellent information-preserving capabilities. Since linear transformations based on sparse random matrices are computationally fast and can be easy to implement using hardware like FPGAs, they are very attractive in the high-speed applications. In addition, we explore explicit constructions of transformation matrices. We show that the generator matrices of primitive BCH codes are good choices, but linear transformations based on such matrices require more computational time due to their high densities.