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A substitution box (S-box) in a symmetric primitive is a mapping $F$ thattakes $k$ binary inputs and whose image is a binary $m$-tuple for some positiveintegers $k$ and $m$, which is usually the only nonlinear element of the mostmodern block ciphers. Therefore, employing S-boxes with good cryptographicproperties to resist various attacks is significant. For power permutation $F$over finite field $\GF{2^k}$, the multiset of  values $\beta_F(1,b)=\#\{x\in \GF{2^k}\midF^{-1}(F(x)+b)+F^{-1}(F(x+1)+b)=1\}$ for $b\in \GF{2^k}$ is called theboomerang spectrum of $F$. The maximum value in the boomerang spectrum iscalled boomerang uniformity. This paper determines the boomerang spectrum ofthe power permutation $X^{2^{3n}+2^{2n}+2^{n}-1}$ over $\GF{2^{4n}}$. Theboomerang uniformity of that power permutation is $3(2^{2n}-2^n)$. However, ona large subset $\{b\in \GF{2^{4n}}\mid \mathbf{Tr}_n^{4n}(b)\neq 0\}$ of$\GF{2^{4n}}$ of cardinality $2^{4n}-2^{3n}$ (where $ \mathbf{Tr}_n^{4n}$ isthe (relative) trace function from $\GF{2^{4n}}$ to $\GF{2^{n}}$), we provethat the studied function $F$ achieves the optimal boomerang uniformity $2$.  It is known that obtaining such functions is a challenging problem.  More importantly, the set of $b$'s giving this value is explicitly determinedfor any value in the boomerang spectrum.

On the Boomerang Spectrum of Power Permutation  $X^{2^{3n}+2^{2n}+2^{n}-1}$ over $\GF{2^{4n}}$ and Extraction of Optimal  Uniformity Boomerang Functions