Minimal weight expansions in Pisot bases

TL;DR

This paper studies minimal weight digit expansions in Pisot base numeration systems, proving automata recognizability and providing explicit forms for special Pisot numbers, with implications for cryptography.

Contribution

It extends minimal weight representation theory to Pisot bases, showing such expansions are automaton-recognizable and explicitly characterizing them for key Pisot numbers.

Findings

Expansions with minimal absolute digit sum are automaton-recognizable.

Explicit automata are provided for the Golden Ratio, Tribonacci, and smallest Pisot number.

Average weight of these expansions is lower than the non-adjacent form.

Abstract

For applications to cryptography, it is important to represent numbers with a small number of non-zero digits (Hamming weight) or with small absolute sum of digits. The problem of finding representations with minimal weight has been solved for integer bases, e.g. by the non-adjacent form in base~2. In this paper, we consider numeration systems with respect to real bases $\beta$ which are Pisot numbers and prove that the expansions with minimal absolute sum of digits are recognizable by finite automata. When $\beta$ is the Golden Ratio, the Tribonacci number or the smallest Pisot number, we determine expansions with minimal number of digits $\pm1$ and give explicitely the finite automata recognizing all these expansions. The average weight is lower than for the non-adjacent form.