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In all algorithms I've seen that rely on pairings-based cryptography (some examples: snarks without PCPs, more snarks, sublinear ring signatures), a common reference string is required. Is this always the case? If so, what is it about pairings (or the algorithms that use them?) that means the CRS is needed?

I know ZCash got around this by using MPC to emulate a TTP as described here, here, and here, but I'm more interested in understanding why the CRS is necessary. To clarify, I don't mean why the party that generates it needs to be trusted, as I understand that the public parameters of the CRS cannot be generated without use of the 'private' component that must later be destroyed, but I would like to know why the public parameters are required?

(I've looked at is pairing based crypto ready for productive use thinking it might answer this question but it doesn't)

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The introductory example for the use of pairings is tripartite Diffie-Hellman key exchange with a single message by Joux.

It requires no trusted party or CRS.

This nature of allowing a DH like interaction between 3 parties is the reason why many of the TTP setups work with pairings. It's used for the third party.

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