-In code this is computed slightly differently, notice that we can define the map $g$ by making $g$ do a circular bit shift left by one position of the higher $N - E + 1$ bits of $n$. This induces the relationships $$ e \circ \text{bitreverse} = \text{bitreverse} \circ g \\ \text{bitreverse} \circ e^{-1} = g^{-1} \circ \text{bitreverse}$$ which are what's used in code to compute $F$ and its inverse (there is no particular reason for this, I found those before having a proof so they stay because they're equivalent and I don't want to fix what's not broken). In the math lingo this means $e$ and $g$ are conjugate via $\text{bitreverse}$.
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