Speeding up Scalar Multiplication on Koblitz Curves Using $\mu _4$ Coordinates

Abstract

Koblitz curves are a special family of binary elliptic curves satisfying equation $y^2+xy=x^3+ax^2+1$, $a\in \{0,1\}$. Scalar multiplication on Koblitz curves can be achieved with point addition and fast Frobenius endomorphism. We show a new point representation system $\mu _4$ coordinates for Koblitz curves. When $a=0$, $\mu _4$ coordinates derive basic group operations—point addition and mixed-addition with complexities $7\mathbf{M}+2\mathbf{S}$ and $6\mathbf{M}+2\mathbf{S}$, respectively. Moreover, Frobenius endomorphism on $\mu _4$ coordinates requires $4\mathbf{S}$. Compared with the state-of-the-art $\lambda$ representation system, the timings obtained using $\mu _4$ coordinates show speed-ups of $28.6\%$ to $32.2\%$ for NAF algorithms, of $13.7\%$ to $20.1\%$ for $\tau$NAF and of $18.4\%$ to $23.1\%$ for regular $\tau$NAF on four NIST-recommended Koblitz curves K-233, K-283, K-409 and K-571.

This work is supported by the National Natural Science Foundation of China (No. 61872442, No. 61802401, No. 61502487) and the National Cryptography Development Fund (No. MMJJ20180216).