Binarization Trees and Random Number Generation
Sung-il Pae

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.
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Taxonomy
TopicsAlgorithms and Data Compression · Computability, Logic, AI Algorithms · semigroups and automata theory
