An Efficient Speeding up Algorithm of Frobenius Based Scalar Multiplication on Koblitz Curves for Cloud Computing

Abstract

With the rapid development of wireless sensor network, cloud computing and Internet of Things, the problems of security and privacy are becoming more and more serious. Elliptic curve cryptography as a public key cryptography plays an important role to solve the security issues, in which scalar multiplication is the most important and time-consuming operation. Koblitz curve is a special class of elliptic curve over binary field, Frobenius endomorphism can be used to accelerate the scalar multiplication. By converting single scalar multiplication into simultaneous multiple scalar multiplication, GLV method can use Straus-Shamir trick to calculate the scalar multiplication. In this paper, we combine the idea of Frobenius endomorphism and GLV method to speed up the scalar multiplication on Koblitz curve. Our algorithm can efficiently convert scalar multiplication into multi-scalar multiplication to reduce the cost of point additions and Frobenius operations. Theoretical analysis results show that: Compared with $$\tau $$ -and-add algorithm, our 2-dimensional implementation provides a speedup over 19%, 3-dimensional implementation speeds up over 29%. Finally, a parallel scalar multiplication algorithm for Koblitz curve is designed, which can flexibly select the dimension of the parallel algorithm based on the number r of processing unit. Compared with the standard $$\tau $$ -and-add algorithm, this algorithm can achieve a speedup of almost r times.