Efficient Scalar Multiplication for Ate Based Pairing over KSS Curve of Embedding Degree 18

Abstract

Efficiency of the next generation pairing based security protocols rely not only on the faster pairing calculation but also on efficient scalar multiplication on higher degree rational points. In this paper we proposed a scalar multiplication technique in the context of Ate based pairing with Kachisa-Schaefer-Scott (KSS) pairing friendly curves with embedding degree \(k = 18\) at the 192-bit security level. From the systematically obtained characteristics p, order r and Frobenious trace t of KSS curve, which is given by certain integer z also known as mother parameter, we exploit the relation \(\#E({\mathbb {F}}_{p}) = p+1-t\) mod r by applying Frobenius mapping with rational point to enhance the scalar multiplication. In addition we proposed z-adic representation of scalar s. In combination of Frobenious mapping with multi-scalar multiplication technique we efficiently calculate scalar multiplication by s. Our proposed method can achieve 3 times or more than 3 times faster scalar multiplication compared to binary scalar multiplication, sliding-window and non-adjacent form method.