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arXiv · 1205.4414

Existence and Optimality of $w$-Non-adjacent Forms with an Algebraic Integer Base

Clemens Heuberger·Daniel Krenn
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Abstract

We consider digital expansions in lattices with endomorphisms acting as base. We focus on the $w$-non-adjacent form ($w$-NAF), where each block of $w$ consecutive digits contains at most one non-zero digit. We prove that for sufficiently large $w$ and an expanding endomorphism, there is a suitable digit set such that each lattice element has an expansion as a $w$-NAF. If the eigenvalues of the endomorphism are large enough and $w$ is sufficiently large, then the $w$-NAF is shown to minimise the weight among all possible expansions of the same lattice element using the same digit system.

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Clemens Heuberger, Daniel Krenn. 2012-05-20. Existence and Optimality of $w$-Non-adjacent Forms with an Algebraic Integer Base. https://doi.org/10.1007/s10474-013-0303-2

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