Question about the benefits of the lattice based construction

Question
#1

The paper says

Our construction stems from a general technical toolkit that we develop to translate pairing-based schemes to lattice-based ones.

Is there any advantage, other than resistance to quantum attack, for a lattice-based construction? (I apologize for the novice question)

#2

I’d say post-quantum security is indeed a big motivation here. But more broadly, I’d say it is interesting to ask what other assumptions can achieve feature parity with pairings.

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