Analysis of the Width-w Non-Adjacent Form in Conjunction with Hyperelliptic Curve Cryptography and with Lattices

TL;DR

This paper provides an asymptotic analysis of digit occurrences in width-w non-adjacent forms within lattices, with applications to hyperelliptic curve cryptography, revealing periodic fluctuations in digit distribution.

Contribution

It introduces a general asymptotic formula for digit counts in lattice-based width-w non-adjacent forms, applicable to cryptographic scalar multiplication methods.

Findings

Main term matches full block length analysis

Second order term shows periodic fluctuation

Results aid in analyzing algorithm running time in cryptography

Abstract

We analyse the number of occurrences of a fixed non-zero digit in the width-w non-adjacent forms of all elements of a lattice in some region (e.g. a ball). Our result is an asymptotic formula, where its main term coincides with the full block length analysis. In its second order term a periodic fluctuation is exhibited. The proof follows Delange's method. This result in a general lattice set-up is then used for numeral systems with an algebraic integer as base. Those come from efficient scalar multiplication methods (Frobenius-and-add methods) in hyperelliptic curves cryptography, and our result is needed for analysing the running time of such algorithms.