An Explicit Formula of Cyclotomic Cubing Available for Pairings on Elliptic Curves with Embedding Degrees of Multiple of Three

Bilinear pairings are widely used for innovative protocols such as ID-based encryption and group signature authentication. According to the current research of the pairings, not only families of pairing-friendly elliptic curves with embedding degrees of multiple of four or six but also that of multiple of three are attractive choices for practical pairings. However, the pairings on such the elliptic curves cannot benefit from an efficient performing squaring available in a cyclotomic subgroup which plays an important role in fast final exponentiation. As one of the candidates of replacements of the squaring, the authors consider an efficient performing cubing available in the cyclotomic subgroup.