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

Binarization Trees and Random Number Generation