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: textit{BW13-P310} and textit{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 textit{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 BLS-446 curve at the same security level.

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