Consideration of Efficient Pairing Applying Two Construction Methods of Extension Fields

In recent years, many innovative cryptography protocols based on the pairing. Finding out an efficient extension field construction is one of the prerequisites for a practical pairing implementation. The author tries to find an elegant extension field construction which will result in efficient pairing over Barreto-Naehrig (BN) curve. In this paper, two construction methods are considered for extension field of degree 12, and one of them results in an efficient exponentiation and the other enables to compute faster Miller loop than the former one. Therefore, a method which uses a basis conversion matrix between the two extension field is proposed. In comparison to the performance of Aranha et al.'s towering, the proposed implementation results in the efficient exponentiation in a multiplicative group, although pairing cost is slightly increased.