Software Implementation of Optimal Pairings on Elliptic Curves with Odd Prime Embedding Degrees

Pairing computations on elliptic curves with odd prime degrees are rarely studied as low efficiency. Recently, Clarisse, Duquesne and Sanders proposed two new curves with odd prime embedding degrees: BW13-P310 and BW19-P286, which are suitable for some special cryptographic schemes. In this paper, we propose efficient methods to compute the optimal ate pairing on this types of curves, instantiated by the BW13-P310 curve. We first extend the technique of lazy reduction into the finite field arithmetic. Then, we present a new method to execute Miller's algorithm. Compared with the standard Miller iteration formulas, the new ones provide a more efficient software implementation of pairing computations. At last, we also give a fast formula to perform the final exponentiation. Our implementation results indicate that it can be computed efficiently, while it is slower than that over the (BLS12-P446) curve at the same security level.