# Optimality of the Width-$w$ Non-adjacent Form: General Characterisation and the Case of Imaginary Quadratic Bases

@article{Heuberger2011OptimalityOT,
title={Optimality of the Width-\$w\$ Non-adjacent Form: General Characterisation and the Case of Imaginary Quadratic Bases},
author={Clemens Heuberger and Daniel Krenn},
journal={arXiv: Number Theory},
year={2011}
}
• Published 5 October 2011
• Mathematics
• arXiv: Number Theory
Efficient scalar multiplication in Abelian groups (which is an important operation in public key cryptography) can be performed using digital expansions. Apart from rational integer bases (double-and-add algorithm), imaginary quadratic integer bases are of interest for elliptic curve cryptography, because the Frobenius endomorphism fulfils a quadratic equation. One strategy for improving the efficiency is to increase the digit set (at the prize of additional precomputations). A common choice is…
12 Citations

## Figures from this paper

### Non-minimality of the width-w non-adjacent form in conjunction with trace one 휏-adic digit expansions and Koblitz curves in characteristic two

• Mathematics
Math. Comput.
• 2018
This article deals with redundant digit expansions with an imaginary quadratic algebraic integer with trace $\pm 1$ as base and a minimal norm representatives digit set. For $w\geq 2$ it is shown

### Existence and optimality of w-non-adjacent forms with an algebraic integer base

• Mathematics
• 2012
We consider digit 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.

### Multi-Base Representations of Integers: Asymptotic Enumeration and Central Limit Theorems

• Mathematics, Computer Science
• 2015

### Arithmetic of Supersingular Koblitz Curves in Characteristic Three

• Mathematics, Computer Science
IACR Cryptol. ePrint Arch.
• 2010
Digital expansions of scalars for supersingular Koblitz curves in characteristic three are considered, allowing for a very simple and efficient precomputation strategy, whereby the rotational symmetry of the digit set is also used to reduce the memory requirements.

### New Minimal Weight Representations for Left-to-Right Window Methods

• Computer Science, Mathematics
CT-RSA
• 2005
This work introduces a new family of radix 2 representations which use the same digits as the w-NAF but have the advantage that they result in a window method which uses less memory.

In the seminal papers [6, 7], Koblitz curves were proposed for cryptographic use. For fast operations on these curves, these papers also initiated a study of the radix-τ expansion of integers in the

### Efficient Arithmetic on Koblitz Curves

• J. Solinas
• Computer Science, Mathematics
Des. Codes Cryptogr.
• 2000
An improved version of theoblitz algorithm, which runs 50 times faster than any previous version, is given, based on a new kind of representation of an integer, analogous to certain kinds of binary expansions.

### Analysis of Alternative Digit Sets for Nonadjacent Representations

• Mathematics
• 2006
Abstract.It is known that every positive integer n can be represented as a finite sum of the form ∑iai2i, where ai ∈ {0, 1,−1} and no two consecutive ai’s are non-zero (“nonadjacent form”, NAF).