Based on what mentioned in the paper "Pairings For Cryptographers" http://www.sciencedirect.com/science/article/pii/S0166218X08000449 the two inputs of a pairing map are two members of two additive groups on elliptic curves, while the output is an element of a multiplicative group on a finite field. enter image description here I would like to know is there any mathematical or technical advantages for a multiplicative finite filed group (in compare with an additive elliptic curve group)? or is there any mathematical or technical disadvantages for an additive elliptic curve group (in compare with multiplicative finite filed group)? The following image shows this question more accurate. enter image description here Please help me enter image description here

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    $\begingroup$ The setting described in the paper is the only we currently know that provides pairings with all required properties. You do not really have a choice (besides which curves and concrete pairing map to use). $\endgroup$ – DrLecter Jul 15 '14 at 8:40
  • $\begingroup$ I replayed you at the end of my question beacause I cannot insert formula here $\endgroup$ – Mohsen Jul 15 '14 at 10:46
  • $\begingroup$ Formulas can be inserted here with common Latex notation, by using '$ $'. $\endgroup$ – tylo Jul 15 '14 at 11:14
  • $\begingroup$ See here for formatting help. $\endgroup$ – Artjom B. Jul 15 '14 at 11:24
  • $\begingroup$ The mathematical disadvantage for $E(F_p)$ as the target group $G_T$ is that no such construction exists. $\endgroup$ – mikeazo Jul 15 '14 at 11:42

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