Paper 2024/038

On Computing the Multidimensional Scalar Multiplication on Elliptic Curves

Abstract

A multidimensional scalar multiplication ($d$-mul) consists of computing $[a_1]P_1+\cdots+[a_d]P_d$, where $d$ is an integer ($d\geq 2)$, $\alpha_1, \cdots, \alpha_d$ are scalars of size $l\in \mathbb{N}^*$ bits, $P_1, P_2, \cdots, P_d$ are points on an elliptic curve $E$. This operation ($d$-mul) is widely used in cryptography, especially in elliptic curve cryptographic algorithms. Several methods in the literature allow to compute the $d$-mul efficiently (e.g., the bucket method~\cite{bernstein2012faster}, the Karabina et al. method~\cite{hutchinson2019constructing, hisil2018d, hutchinson2020new}). This paper aims to present and compare the most recent and efficient methods in the literature for computing the $d$-mul operation in terms of with, complexity, memory consumption, and proprieties. We will also present our work on the progress of the optimisation of $d$-mul in two methods. The first method is useful if $2^d-1$ points of $E$ can be stored. It is based on a simple precomputation function. The second method works efficiently when $d$ is large and $2^d-1$ points of $E$ can not be stored. It performs the calculation on the fly without any precomputation. We show that the main operation of our first method is $100(1-\frac{1}{d})\%$ more efficient than that of previous works, while our second exhibits a $50\%$ improvement in efficiency. These improvements will be substantiated by assessing the number of operations and practical implementation.