Binarization Trees and Random Number Generation

TL;DR

This paper explores how binary trees can be used to create complete binarizations for extracting unbiased random bits from loaded dice, leveraging entropy-preserving structures and the leaf entropy theorem.

Contribution

It demonstrates the abundance of complete binarizations derived from binary trees with m leaves, expanding the understanding of entropy-preserving binarizations.

Findings

Complete binarizations can be naturally obtained from binary trees with m leaves.

The leaf entropy theorem is instrumental in establishing the properties of these binarizations.

The structure lemma supports the theoretical framework for binarization completeness.

Abstract

An m-extracting procedure produces unbiased random bits from a loaded dice with m faces. A binarization takes inputs from an m-faced dice and produce bit sequences to be fed into a (binary) extracting procedure to obtain random bits. Thus, binary extracting procedures give rise to an m-extracting procedure via a binarization. An entropy- preserving binarization is to be called complete, and such a procedure has been proposed by Zhou and Bruck. We show that there exist complete binarizations in abundance as naturally arising from binary trees with m leaves. The well-known leaf entropy theorem and a closely related structure lemma play important roles in the arguments.