# Is it hard to determine an automorphism when the mapped value by several compositions of the automorphism is given

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.