Minimal weight expansions in Pisot bases

Christiane Frougny (1) , Wolfgang Steiner (1)

1 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications

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 a real base $\beta$ which is a Pisot number. 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 finite automata recognizing all these expansions. The average weight is lower than for the non-adjacent form. In the general case of a base $\beta$ which is a Pisot number satisfying a certain condition (D$'$), we prove that the expansions with minimal absolute sum of digits are recognizable by a finite automaton.

Mots clés: minimal weight; beta-expansions; Pisot numbers; Fibonacci numbers; automata

Domaines: Mathématique discrète [cs.DM]; Théorie des nombres [math.NT]; Cryptographie et sécurité [cs.CR]

Christiane Frougny, Wolfgang Steiner. Minimal weight expansions in Pisot bases. 2008. ⟨hal-00265661v2⟩