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Let's say there is a bilinear pairing $G \times G \rightarrow G_t$ (e.g., for bn128), and let prime $q$ be the order of $G$. Is it possible to find a prime order group over integers such that its order $p | q-1$ (that is $q$ is used as the modulus for the "smaller" group)? I checked the order of bn128 and bs381 groups, it seems that the $q-1$ has no large prime factors.

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