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I had asked one question before One-More Computational Diffie-Hellman in asymmetric pairing groups and have not received answer. I am posing a supplementary question now that I just realized I don't have even idea what to call the building block problem.

That is in an asymmetric pairing groups $e:G_1 \times G_2 \to G_T$, we have $X=g^x$ where $g$ is a generator of $G_1$ and $S$ is a random element in $G_2$, The problem is to find $S^x$ given $X$ and $S$. It looks like a computational Diffie-Hellman problem but is not really so as $X$ and $S$ not from the same group. The Wikipedia BLS https://en.wikipedia.org/wiki/BLS_digital_signature defines it in GAP-DH groups but many implementations that I have found actually uses BN254, an asymmetric pairing group pair. Clearly they are not really the same.

My question is has this problem been defined in the existing literature ? Has there been security proofs? And my original question was has "One-More" version of this problem been defined where $X$ is published and $S$ is queried by the adversary

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OK guys I found the answer. It is computational co Diffie-Hellman problem. co-CDH. I am editing my previous question to reflect this.

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