Efficient computation of Tate pairing in projective coordinate over general characteristic fields

We consider the use of Jacobian coordinates for Tate pairing over general characteristics. The idea of encapsulated double-and-line computation and add-and-line computation has been introduced. We also describe the encapsulated version of iterated doubling. Detailed algorithms are presented in each case and memory requirement has been considered. The inherent parallelism in each of the algorithms have been identified leading to optimal two-multiplier algorithm. The cost comparison of our algorithm with previously best known algorithms shows an efficiency improvement of around 33% in the general case and an efficiency improvement of 20% for the case of the curve parameter a = -3.