Saving fractional bits: A practical entropy efficient code for fair die rolls

TL;DR

This paper presents a practical algorithm that efficiently simulates fair die rolls using minimal entropy, reusing entropy stored in a register to reduce waste and adapt to different die sizes.

Contribution

It introduces a novel entropy-efficient algorithm that reuses entropy during die simulations, minimizing entropy loss and allowing dynamic adjustment of die sides.

Findings

Entropy loss decreases as register size increases

The algorithm achieves near-zero entropy waste per roll

It supports changing the number of die sides dynamically

Abstract

We give an implementation of an algorithm that uses fair coin flips to simulate fair rolls of an $n$-sided die. A register plays the role of an entropy pool and holds entropy that is generated as a by-product during each die roll and that is usually discarded. The entropy stored in this register is completely reused during the next rolls. Consequently, we can achieve an almost negligible loss of entropy per roll. The algorithm allows to change the number of sides of the die in each round. We prove that the entropy loss is monotone decreasing with increasing entropy pool size (register length).