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Given $$e(g, d) = c $$ where,

  • $e$ is bilinear pairing function chosen by the user/attacker,
  • the values of $g$ and $c$ are known
  • $g, d ∈ \mathbb{G}_1$ , $c$ depends upon the $e$

can we somehow compute the value of $d$

In a high level, I am asking if there exists a function $f$ that, in one way is the inverse of a $e$ can be used to compute $d$?

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What you describe is the so called pairing inversion problem. You may look here for some pointers to literature. –  DrLecter Feb 6 at 10:28
    
Thanks, the blog was quite interesting, and I am still going through it (for the second time). In the mean while, could I naively ask, if there is a "fault", or "a cheat", to compute the inverse of a pairing "quickly"? Say, I am looking from an attackers point of view, so an existential "cheat" would do fine as well. –  Subhayan Feb 6 at 21:03
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