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Much attention has been given to the efficient computation of pairings onelliptic curves with even embedding degree since the advent of pairing-basedcryptography. The few existing works in the case of odd embedding degreesrequire some improvements. This paper considers the computation of optimal atepairings on elliptic curves of embedding degrees $k=9$, $15$, $27$ which havetwists of order three. Our main goal is to provide a detailed arithmetic andcost estimation of operations in the tower extensions field of thecorresponding extension fields. A good selection of parameters enables us toimprove the theoretical cost for the Miller step and the final exponentiationusing the lattice-based method as compared to the previous few works that existin these cases. In particular, for $k=15$, $k=27$, we obtain an improvement, interms of operations in the base field, of up to 25% and 29% respectively in thecomputation of the final exponentiation. We also find that elliptic curves withembedding degree $k=15$ present faster results than BN12 curves at the 128-bitsecurity level. We provide a MAGMA implementation in each case to ensure thecorrectness of the formulas used in this work.

Optimal Ate Pairing on Elliptic Curves with Embedding Degree $9,15$ and $27$