Efficient simultaneous inversion in parallel and application to point multiplication in ECC

Abstract

Inversion is the costliest of all finite field operations. Some algorithms require computation of several finite field elements simultaneously (elliptic curve factorization for example). Montgomery's trick is a well known technique for performing the same in a sequential set up with little scope for parallelization. In the current work we propose an algorithm which needs almost same computational resources as Montgomery's trick, but can be easily parallelized. Our algorithm uses binary tree structures for computation and using 2r-1 multipliers, it can simultaneously invert 2r elements in 2r multiplication rounds and one inversion round. We also describe how the algorithm can be used when 2, 4, ... number of multipliers are available. To exhibit the utility of the method, we apply it to obtain a parallel algorithm for elliptic curve point multiplication. The proposed method is immune to side-channel attacks and compares favourably to many parallel algorithms existing in literature. © Springer-Verlag Berlin Heidelberg 2005.