I am currently reading a research paper that introduces this thing called a butterfly structure.
I am confused about the structure of the open butterfly in this paper which has the equation $H^{\alpha}_{e}(x,y)=(R^{-1}_{R_y[e,\alpha](x)}(y),R_y[e,\alpha](x))$ where $R_k[e,\alpha](x)=(x+\alpha k)^e+k^e$ and a corresponding structure as seen below:
I am confused as to how the structure of the butterfly above, relates to the equation. Furthermore, I don't really understand what $x^{1/e}$ in the butterfly structure means. It would be great if someone could explain this. The paper is Cryptanalysis of a Theorem Decomposing the only Known solution to the big APN problem.