Extension Field for Xate Pairing with Freeman Curve

In recent years, pairing-based cryptographies such as ID-based cryptography [1] and group signature [2] have been studied. For their implementations, pairings such as Weil pairing [1], Tate pairing, Ate pairing [3] and Xate pairing [4] have been used. In order to implement these pairings, several kinds of ordinary pairing-friendly curves such as Miyaji-Nakabayashi-Takano (MNT) curve [5], Barreto-Naehrig (BN) curve [6] and Freeman curve [7, 8] have been proposed. As the definition field of these curves, many researchers use optimal extension field (OEF) [9] because OEF carries out arithmetic operations efficiently. However, it is known that OEF is not available for the definition field of Freeman curve due to the condition of OEF. Our previous work namely Type I-X all one polynomial field (AOPF) [11] is available for the definition field of Freeman curve. Type I-X AOPF can carry out arithmetic operations as efficient as OEF. In this paper, the authors consider how to constructed type I-X AOPF and optimize a multiplication algorithm for Xate pairing with Freeman curve. Additionally, this paper shows some experimental results of Xate pairing with Freeman curve defined over the type