Much attention has been given to the efficient computation of pairings on
elliptic curves with even embedding degree since the advent of pairing-based
cryptography. The few existing works in the case of odd embedding degrees
require some improvements. This paper considers the computation of optimal ate
pairings on elliptic curves of embedding degrees $k=9$, $15$, $27$ which have
twists of order three. Our main goal is to provide a detailed arithmetic and
cost estimation of operations in the tower extensions field of the
corresponding extension fields. A good selection of parameters enables us to
improve the theoretical cost for the Miller step and the final exponentiation
using the lattice-based method as compared to the previous few works that exist
in these cases. In particular, for $k=15$, $k=27$, we obtain an improvement, in
terms of operations in the base field, of up to 25% and 29% respectively in the
computation of the final exponentiation. We also find that elliptic curves with
embedding degree $k=15$ present faster results than BN12 curves at the 128-bit
security level. We provide a MAGMA implementation in each case to ensure the
correctness of the formulas used in this work.
Comment: 25 pages
Computing the Ate Pairing on Elliptic Curves with Embedding Degree k = 9