Minimal weight expansions in Pisot bases

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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 $β$ which are Pisot numbers and prove that the expansions with minimal absolute sum of digits are recognizable by finite automata. When $β$ 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.

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Discrete Mathematics, Cryptography and Security, Number Theory

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http://salesiana.dossiersoluciones.com/handle/123456789/215172

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/0803.2874
 http://arxiv.org/abs/0803.2874

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arXiv