Speeding Up Scalar Multiplication On Koblitz Curves Using Mu(4) Coordinates

Koblitz curves are a special family of binary elliptic curves satisfying equation y(2) + xy = x(3) + ax(2) + 1, a is an element of {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 7M+ 2S and 6M+ 2S, respectively. Moreover, Frobenius endomorphism on mu(4) coordinates requires 4S. Compared with the stateof-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.