The pairing is a mathematical notion wich appeared in cryptography during the 80'. At the beginning, it
was used to build attacks on cryptosystems, transferring the discrete logarithm problem on elliptic curves,
to a discrete logarithm problem on finite fields, the first was the MOV36 attack in 1993. Now, pairings
are used to construct some cryptographic protocols: Diffie Hellman tripartite, identity based encryption, or
short signature. The main two pairings usually used are the Tate and Weil pairings. They use distortions
and rationnal functions, and their complexities depends of the curve and the field involved.
This study deals with two particular papers: one due to N. Koblitz and A. Menezes27 published in 2005,
and a second one written by R Granger, D. Page and N. Smart24 in 2006. These two papers compare Tate
andWeil pairings, but they differ in their conclusions. We consider the different arithmetic tricks used, trying
to precise each point, in a way to avoid any ambiguity. Thus, the arithmetics proposed take into account
the features of the fields and the curves used. We clarify the complexity of the possible implementations.
We compare the different approaches, in order to clarify the conclusions of the previous papers.