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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: Structure of an open bijective butterfly structure

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.

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Squeamish Ossifrage is correct. This is the power map $x\rightarrow x^t$ where $te\equiv 1 \pmod{2^n-1}$ and the underlying field is $GF(2^n).$

The subject of power maps and when they give permutations with nice properties is quite well studied, and this is a paper along those lines.

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