# 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} }

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…

## Figures from this paper

## 12 Citations

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

- MathematicsMath. 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…

### Analysis of the width-w non-adjacent form in conjunction with hyperelliptic curve cryptography and with lattices

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 2013

### 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

This paper provides a general asymptotic formula for the number of multi-base representations of a positive integer $n, and proves central limit theorems for the sum of digits, the Hamming weight and the occurrences of a fixed digits in a random representation.

### On the Number of Multi-Base Representations of an Integer

- Mathematics, Computer Science
- 2014

This work provides a general asymptotic formula for the number of multi-base representations of a positive integer n and proves central limit theorems for the sum of digits and the Hamming weight of a random representation.

### Some properties of $${\tau }$$τ-adic expansions on hyperelliptic Koblitz curves

- Mathematics, Computer ScienceArXiv
- 2014

Investigation shows that the τ-adic sparse expansion has only the existence and theτ-NAF has the exist and uniqueness, and these results guarantee the concrete cryptographic implementations of these generalizations.

### On the minimal Hamming weight of a multi-base representation

- Computer Science, MathematicsJournal of Number Theory
- 2020

### MULTI-BASE REPRESENTATIONS OF INTEGERS:

- Mathematics, Computer Science
- 2015

This work provides a general asymptotic formula for the number of multi-base representations of a positive integer, and proves central limit theorems for different statistics associated to a multi- base representation.

### Constructions with Block Diagonal Bases

- Mathematics, Computer Science
- 2014

It is proved that diagonal extensions can always be performed even if the basic blocks are not number systems, and that except 43 cases the Gaussian integers are always able to serve as basic blocks for simultaneous number systems using dense digit sets.

### Number System Constructions with Block Diagonal Bases (Numeration and Substitution 2012)

- Mathematics, Computer Science
- 2014

It is proved that diagonal extensions can always be performed even if the basic blocks are not number systems, and that except 43 cases the Gaussian integers are always able to serve as basic blocks for simultaneous number systems using dense digit sets.

## References

SHOWING 1-10 OF 31 REFERENCES

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

- Mathematics, Computer Science
- 2013

### A Note on the Signed Sliding Window Integer Recoding and a Left-to-Right Analogue

- Computer ScienceSelected Areas in Cryptography
- 2004

It is proved that the ω-NAF is a redundant radix-2 recoding of smallest weight among all those with integral coefficients smaller in absolute value than 2ω−1, and it is shown that the two recodings have the same (optimal) weight.

### Minimality of the Hamming Weight of the \tau-NAF for Koblitz Curves and Improved Combination with Point Halving

- Mathematics, Computer ScienceIACR Cryptol. ePrint Arch.
- 2005

The new representation, called the wide-double-NAF, is not only simpler to compute, but it is also optimal in a suitable sense, it has minimal Hamming weight among all τ-adic expansions with digits {0,±1} that allow one halving to be inserted in the corresponding scalar multiplication algorithm.

### Redundant τ-Adic Expansions II: Non-Optimality and Chaotic Behaviour

- Computer ScienceMath. Comput. Sci.
- 2010

It is shown that there is no longer an online algorithm to compute an optimal expansion from the digits of some standard Expansion from the least to the most significant digit, which can be interpreted as chaotic behaviour.

### Nonadjacent Radix-τ Expansions of Integers in Euclidean Imaginary Quadratic Number Fields

- MathematicsCanadian Journal of Mathematics
- 2008

Abstract In his seminal papers, Koblitz proposed curves for cryptographic use. For fast operations on these curves, these papers also initiated a study of the radix- $\tau $ expansion of integers in…

### Arithmetic of Supersingular Koblitz Curves in Characteristic Three

- Mathematics, Computer ScienceIACR 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, MathematicsCT-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.

### NONADJACENT RADIX-τ EXPANSIONS OF INTEGERS IN EUCLIDEAN IMAGINARY QUADRATIC NUMBER FIELDS

- Mathematics
- 2005

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

- Computer Science, MathematicsDes. 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).…