In the case of Barreto-Naehrig pairing-friendly curves of embedding degree 12 of order
, recent efficient Ate pairings such as R-ate, optimal, and Xate pairings achieve Miller loop lengths of
$(1/4)\lfloor \log_2 r\rfloor$
. On the other hand, the twisted Ate pairing requires
$(3/4) \lfloor \log_2 r\rfloor$
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 proposal 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
$(1/4) \lfloor \log_2 r \rfloor$
attained by the most efficient Ate pairings.