Sole Inversion Precomputation for Elliptic Curve Scalar Multiplications

Abstract

This paper presents a new approach to precompute points [3]P, [5]P, ..., [2k-1]P, for some k ≥ 2 on an elliptic curve over $\mathbb{F}_p$. Those points are required for the efficient evaluation of a scalar multiplication, the most important operation in elliptic curve cryptography. The proposed method precomputes the points in affine coordinates and needs only one single field inversion for the computation. The new method is superior to all known methods that also use one field inversion, if the required memory is taken into consideration. Compared to methods that require several field inversions for the precomputation, the proposed method is faster for a broad range of ratios of field inversions and field multiplications. The proposed method benefits especially from ratios as they occur on smart cards.