Hybrid Binary-Ternary Joint 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 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 19%.