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I am trying to find a reduction for the following DLOG problem in generic groups. It is a simple generalization but I'm not finding any reference (the closest being the Chaum-Pedersen signature scheme sec 3.2, and BLS signatures without the hashing).

Let $G$ be a cyclic group, and $g, h$ generators. The problem is to find $y$ given $g^y, h^y$.

Looking for any insight or reference.

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    $\begingroup$ Given $g$ and challenge Choose random $r$ such that $g^r$ is a generator. (Easy in prime order groups.) Set $h:=g^r$ and let $h^y := (g^y)^r$. $\endgroup$
    – Maeher
    May 15 at 9:49
  • $\begingroup$ OK, but how does that help the attacker in finding $y$ (or am I misreading your argument) EDIT: Oh, I got it, thanks. $\endgroup$ May 15 at 10:13
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    $\begingroup$ It doesn't. That's the point. It's a short description of the reduction from dlog to your problem. This reduction (described in more detail in Daniel S' answer below) shows that your problem is no easier than dlog. $\endgroup$
    – Maeher
    May 15 at 10:17

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This is equivalent to the discrete logarithm problem.

Clearly if I can solve the discrete logarithm problem, I can solve your problem.

Given an instance of the discrete logarithm problem e.g. given $x=g^y$ find $y$, I can generate an instance of your problem by choosing a random $r$ coprime to the group order and setting $h=g^r$. In this case I can also compute $x^r$ which will be $h^r$.

Thus if I can solve your problem, I can solve the discrete logarithm problem for the group.

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  • $\begingroup$ This assumes that the attacker can select $h$. $\endgroup$ May 15 at 10:19
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    $\begingroup$ No, the randomisation of $r$ gives a random instance of your problem conditioned on one of the generators being $g$. $\endgroup$
    – Daniel S
    May 15 at 10:26

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