In the case of Barreto-Naehrig pairing-friendly curves of embedding degree 12 of order r, recent efficient Ate pairings such as R-ate, optimal, and Xate pairings achieve Miller loop lengths of(1/4) ⌊log2 r⌋. On the other hand, the twisted Ate pairing requires (3/4) ⌊log2 r⌋ loop iterations, and thus is usually slower than the recent efficient Ate pairings. This paper proposes an improved twisted Ate pairing using Frobenius maps and a small scalar multiplication. The proposed idea splits the Miller's algorithm calculation into several independent parts, for which multi-pairing techniques apply efficiently. The maximum number of loop iterations in Miller's algorithm for the proposed twisted Ate pairing is equal to the (1/4) ⌊log2 r ⌋ attained by the most efficient Ate pairings.

We observe a natural generalisation of the ate and twisted ate pairings, which allow for performance improvements in non standard applications of pairings to cryptography like composite group orders. We also give a performance comparison of our pairings and the Tate, ate and twisted ate pairings for certain polynomial families based on operation count estimations and on an implementation, showing that our pairings can achieve a speedup of a factor of up to two over the other pairings.