Fast arithmetic for characteristic three finite fields is desirable in pairing-based cryptography because there is a suitable family of elliptic curves over having embedding degree 6. In this paper we present some structure results for Gaussian normal bases of, and use the results to devise faster multiplication algorithms. We carefully compare multiplication in using polynomial bases and Gaussian normal bases. Finally, we compare the speed of encryption and decryption for the Boneh-Franklin and Sakai-Kasahara identity-based encryption schemes at the 128-bit security level, in the case where supersingular elliptic curves with embedding degrees 2, 4 and 6 are employed.