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Generalization of the Discrete logarithm problem to non-abelian groups is discussed by many authors. One of the generalizations is shown in MOR cryptosystem as in the below link, by considering the inner automorphisms.

https://arxiv.org/pdf/cs/0607011.pdf

So if group $G$ is generated by elements $\{\gamma_1, \gamma_2, \cdots, \gamma_s\}$, a communicating party, say Alice makes $\{\phi_g(\gamma_i)\}_{i=1}^{s}$ and $\{\phi_g^r(\gamma_i)\}_{i=1}^{s}$ public for an inner automorphism $\phi_g$ and a private key $r$. By the generalized discrete logarith problem, it is hard to compute $r$. To use this method, they have to use a suitable non-abelian group where the generalized discrete logarithm problem is proven to be hard.

However, suppose I have a group of automorphisms, and for an automorphism $f$, I need to make public only $f^m(t_i)$, for a private key value $m$ and generating elements $t_i$'s of a related group. Then since I'm not revealing $f(t_i)$, anyway determining $m$ nor the original automorphism $f$ is a hard problem right? Even if it is not in the format of the generalized discrete logarithm problem?

According to my situation I don't need to reveal $f(t_i)$. But I need to clarify when I only reveal $f^m(t_i)$, it is impossible to find $f$ nor $m$ whatever the group is.

Please be kind enough to advice and please mention if I need to include more explanations/any unclear part.

Thanks a lot in advance.

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