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How to calculate the inversion fuction S:S:mathbb{F}_{2^n}rightarrow mathbb{F}_{2^n},with S(x)=x^{-1}

The S-box is defined as the generalised inverse function $S:mathbb{F}_{2^n}rightarrow mathbb{F}_{2^n}$,in quotient ring $mathcal{R}:=mathbb{F}_{2^n}[X]/(X^{2^n}-X)$ with $S(x)=x^{-1}$, is correct $S(X):=X^{2^n-2}$. But the Euler’s theorem says $x^{varphi(n)}equiv1pmod{n}$,so the answer is $x^{varphi(n)-1}=x^{2^{n-1}-1}equiv x^{-1}pmod{n}$,why is $S(X):=X^{2^n-2}$

The S-box is defined as the generalised inverse function $S:mathbb{F}_{2^n}rightarrow mathbb{F}_{2^n}$,in quotient ring $mathcal{R}:=mathbb{F}_{2^n}[X]/(X^{2^n}-X)$
with $S(x)=x^{-1}$, is correct $S(X):=X^{2^n-2}$.
But the Euler’s theorem says $x^{varphi(n)}equiv1pmod{n}$,so the answer is $x^{varphi(n)-1}=x^{2^{n-1}-1}equiv x^{-1}pmod{n}$,why is $S(X):=X^{2^n-2}$