A Technique for Fast Miller’s Algorithm of Ate 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 can realize efficient pairings. However, the range of the practical choices of the elliptic curves with embedding degrees of multiple of three is more restricted than that of even embedding degrees by an efficiency reason for the computation of Miller's algorithm with a signed binary representation of a loop parameter. To ease the restriction, the authors propose to compute the Miller's algorithm by swapping the sign of the loop parameter without performance degradation for the ate pairing on such the elliptic curves.