Optimal Representation for Right-to-Left Parallel Scalar Point Multiplication

TL;DR

This paper develops an optimal number representation for right-to-left parallel scalar multiplication on elliptic curves, demonstrating that traditional methods like NAF are nearly optimal with minimal potential improvements.

Contribution

It introduces a mathematical model for right-to-left parallel scalar multiplication and proposes algorithms to find near-optimal representations, showing NAF's near-optimality.

Findings

NAF representation is nearly optimal for right-to-left parallel scalar multiplication.

The proposed algorithms can generate representations close to the minimal computation time.

Parallel computation time cannot be improved by more than 1% over NAF.

Abstract

This paper introduces an optimal representation for a right-to-left parallel elliptic curve scalar point multiplication. The right-to-left approach is easier to parallelize than the conventional left-to-right approach. However, unlike the left-to-right approach, there is still no work considering number representations for the right-to-left parallel calculation. By simplifying the implementation by Robert, we devise a mathematical model to capture the computation time of the calculation. Then, for any arbitrary amount of doubling time and addition time, we propose algorithms to generate representations which minimize the time in that model. As a result, we can show a negative result that a conventional representation like NAF is almost optimal. The parallel computation time obtained from any representation cannot be better than NAF by more than 1%.