Emmanuel Fouotsa ; Nadia El Mrabet ; Aminatou Pecha - Optimal Ate Pairing on Elliptic Curves with Embedding Degree $9,15$ and $27$

gcc:6167 - journal of Groups, complexity, cryptology, April 17, 2020, Volume 12, Issue 1 - https://doi.org/10.46298/jgcc.2020.12.1.6167
Optimal Ate Pairing on Elliptic Curves with Embedding Degree $9,15$ and $27$

Authors: Emmanuel Fouotsa ; Nadia El Mrabet ; Aminatou Pecha

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.

Volume: Volume 12, Issue 1
Published on: April 17, 2020
Accepted on: April 17, 2020
Submitted on: February 28, 2020
Keywords: Mathematics - Algebraic Geometry,Mathematics - Number Theory


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