Analysis of Width-$w$ Non-Adjacent Forms to Imaginary Quadratic Bases

TL;DR

This paper analyzes the statistical properties of width-$w$ non-adjacent forms (NAFs) in algebraic integer bases, providing explicit formulas, asymptotic behavior, and a central limit theorem, with applications to elliptic curve cryptography.

Contribution

It offers a detailed analysis of digit occurrence in width-$w$ NAFs over imaginary quadratic bases, including explicit expectation, variance, and asymptotic formulas, extending understanding of their structure and distribution.

Findings

Explicit expectation and variance formulas for digit occurrences.

Asymptotic formulas with periodic fluctuations for imaginary quadratic bases.

Proof of a central limit theorem for digit distribution in width-$w$ NAFs.

Abstract

We consider digital expansions to the base of $\tau$, where $\tau$ is an algebraic integer. For a $w \geq 2$, the set of admissible digits consists of 0 and one representative of every residue class modulo $\tau^w$ which is not divisible by $\tau$. The resulting redundancy is avoided by imposing the width $w$-NAF condition, i.e., in an expansion every block of $w$ consecutive digits contains at most one non-zero digit. Such constructs can be efficiently used in elliptic curve cryptography in conjunction with Koblitz curves. The present work deals with analysing the number of occurrences of a fixed non-zero digit. In the general setting, we study all $w$-NAFs of given length of the expansion. We give an explicit expression for the expectation and the variance of the occurrence of such a digit in all expansions. Further a central limit theorem is proved. In the case of an imaginary quadratic $\tau$ and the digit set of minimal norm representatives, the analysis is much more refined: We give an asymptotic formula for the number of occurrence of a digit in the $w$-NAFs of all elements of $\Z[\tau]$ in some region (e.g. a disc). The main term coincides with the full block length analysis, but a periodic fluctuation in the second order term is also exhibited. The proof follows Delange's method. We also show that in the case of imaginary quadratic $\tau$ and $w \geq 2$, the digit set of minimal norm representatives leads to $w$-NAFs for \emph{all} elements of $\Z[\tau]$. Additionally some properties of the fundamental domain are stated.