Fast Multiple Point Multiplication on Elliptic Curves over Prime and Binary Fields using the Double-Base Number System
Multiple-point multiplication on elliptic curves is the highest computational complex operation in the elliptic curve cyptographic based digital signature schemes. We describe three algorithms for multiple-point multiplication on elliptic curves over prime and binary fields, based on the representations of two scalars, as sums of mixed powers of 2 and 3. Our approaches include sliding window mechanism and some precomputed values of points on the curve. A proof for formulae to calculate the number of double-based elements, doublings and triplings below 2^n is listed. Affine coordinates and Jacobian coordinates are considered in both prime fields and binary fields. We have achieved upto 24% of improvements in new algorithms for multiple-point multiplication.
Hybrid Binary-Ternary Joint Sparse Form and its Application in Elliptic Curve Cryptography
Multi-exponentiation is a common and time consuming operation in public-key cryptography. Its elliptic curve counterpart, called multi-scalar multiplication is extensively used for digital signature verification. Several algorithms have been proposed to speed-up those critical computations. They are based on simultaneously recoding a set of integers in order to minimize the number of general multiplications or point additions. When signed-digit recoding techniques can be used, as in the world of elliptic curves, Joint Sparse Form (JSF) and interleaving $w$-NAF are the most efficient algorithms. In this paper, a novel recoding algorithm for a pair of integers is proposed, based on a decomposition that mixes powers of 2 and powers of 3. The so-called Hybrid Binary-Ternary Joint Sparse Form require fewer digits and is sparser than the JSF and the interleaving $w$-NAF. Its advantages are illustrated for elliptic curve double-scalar multiplication; the operation counts show a gain of up to 18\%.