Memory-Efficient Algorithm for Scalar Multiplications on Twisted Edwards Curves for Isogeny-Based Cryptosystems

Abstract

Scalar multiplications are considered an essential aspect of implementations of isogeny-based cryptography. The efficiency of scalar multiplication depends on the equation of the underlying elliptic curves and the addition chain employed. Bos and Friedberger recently stated that, for larger scalar multiplication, addition-subtraction chains will become more useful for twisted Edwards curves because of the differential restriction on Montgomery curves in the setting of isogeny-based cryptosystem. Motivated by these comments, we attempt to increase the efficiency of scalar multiplication in twisted Edwards curves in terms of the memory of algorithms. In this paper, we present a double-base addition-subtraction chain algorithm with memory efficiency for scalar multiplication. The memory usage of this part is $O\left({\left(\log n\right)}^2/\log \log n\right)$ , which is better than the result of the tree-based approach, which is $O\left({\left(\log n\right)}^2\right)$ .