What does $x^{1/e}$ mean in cryptography in this open butterfly structure?
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.