Although identity-based cryptography offers a number of functional advantages over conventional public key methods, the computational costs are significantly greater. The dominant part of this cost is the Tate pairing, which, in characteristic three, is best computed using the algorithm of Duursma and Lee. However, in hardware and constrained environments, this algorithm is unattractive since it requires online computation of cube roots or enough storage space to precompute required results. We examine the use of normal basis arithmetic in characteristic three in an attempt to get the best of both worlds: an efficient method for computing the Tate pairing that requires no precomputation and that may also be implemented in hardware to accelerate devices such as smart-cards.