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That depends mainly on the curves used for the pairing and the pairing itself. As a rough estimation, I would say an Ate pairing over a Baretto Naehrig curve costs 10 times the effort of a scalar multiplication in that curve.


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You can see all these benchmark results by implementing the functions. Thankfully other people have done this. If you use python then charm framework is a good start to check pairing computations. You can also try to benchmark the pbc library for pairings. In my pc a pairing is computed for less than 5ms for MNT curves (159,201,224) For modular arithmetics ...


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You ask if it's possible: yes it is. But you have to ask yourself if it's feasible in practice. This introduces the notion of complexity. The method you mention is the naive brute force calculation for the discrete logarithm problem and its complexity is $O(n)$ where $n$ is the order of the elliptic curve. Several algorithms are more efficients like ...


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the parameters define which curve is used, which has consequences on security and efficiency. To decide which curve type you need, go to section 8 of the PBC manual and read the paper "pairings for cryptographers" then PBC provides functions to create your curve (i.e. param files): https://crypto.stanford.edu/pbc/manual/ch05s01.html But you can skip this ...


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The pairing description (type, q, h, r, etc.) is the definition of the field that the elliptic curve of some type operates on. The actual curve is identified by its type and is baked into the framework (e.g. PBC) you're using. This definition corresponds to $q$ (group order, same as q), $\mathbb{G}, \mathbb{G}_T$ (groups defined by pairing description and ...


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The server must give the clients all the necessary information for the clients to be able to compute the pairing. This will differ depending on how pairings are implemented in the system, and what prior knowledge the clients have.



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