Ultimately customized multiplication algorithm in the extension field for Xate and R-ate Pairing with Freeman curve

Recently, pairing-based cryptographies such as ID-based cryptography and group signature have been studied. For their implementations, pairings such as Xate pairing and R-ate pairing are efficient. In order to make pairing calculations faster, it is important to make not only pairing algorithms but also arithmetic operations in the definition field efficient. Thus, in several kinds of ordinary pairing-friendly curves, this paper focuses on Freeman curve that enables fast pairing-based cryptographies. This paper proposes ultimately an customized multiplication algorithm in the extension field for Freeman curve, and then shows some implementation results of Xate pairing and R-ate pairing with Freeman curve.