A New Encoding Algorithm for a Multidimensional Version of the Montgomery Ladder

We propose a new encoding algorithm for the simultaneous differential multidimensional scalar point multiplication algorithm d-MUL. Previous encoding algorithms are known to have major drawbacks in their efficient and secure implementation. Some of these drawbacks have been avoided in a recent paper in 2018 at a cost of losing the general functionality of the point multiplication algorithm. In this paper, we address these issues. Our new encoding algorithm takes the binary representations of scalars as input, and constructs a compact binary sequence and a permutation, which explicitly determines a regular sequence of group operations to be performed in d-MUL. Our algorithm simply slides windows of size two over the scalars and it is very efficient. As a result, while preserving the full generality of d-MUL, we successfully eliminate the recursive integer matrix computations in the originally proposed encoding algorithms. We also expect that our new encoding algorithm will make it easier to implement d-MUL in constant time. Our results can be seen as the efficient and full generalization of the one dimensional Montgomery ladder to arbitrary dimension.