Scalar Multiplication Using Frobenius Expansion over Twisted Elliptic Curve for Ate Pairing Based Cryptography. - Yasuyuki Nogami; Yumi Sakemi; Takumi Okimoto; Kenta Nekado; Masataka Akane; Yoshitaka Morikawa (IEICE Trans. Fundam. Electron. Commun. Comput. Sci., 2009)