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

A substitution box (S-box) in a symmetric primitive is a mapping $F$ that takes $k$ binary inputs and whose image is a binary $m$-tuple for some positive integers $k$ and $m$, which is usually the only nonlinear element of the most modern block ciphers. Therefore, employing S-boxes with good cryptographic properties to resist various attacks is significant. For power permutation $F$ over finite field $\text{GF}{2}^k$, the multiset of values ${\beta }_F(1,b)=\mathrm{\#}\{x\in \text{GF}{2}^k\mid F^{-1}(F(x)+b)+F^{-1}(F(x+1)+b)=1\}$ for $b\in \text{GF}{2}^k$ is called the boomerang spectrum of $F$. The maximum value in the boomerang spectrum is called boomerang uniformity. This paper determines the boomerang spectrum of the power permutation $X^{2^{3n}+2^{2n}+2^n-1}$ over $\text{GF}{2}^{4n}$. The boomerang uniformity of that power permutation is $3(2^{2n}-2^n)$. However, on a large subset $\{b\in \text{GF}{2}^{4n}\mid {\mathbf{T}\mathbf{r}}_n^{4n}(b)\mathrm{\ne }0\}$ of $\text{GF}{2}^{4n}$ of cardinality $2^{4n}-2^{3n}$ (where ${\mathbf{T}\mathbf{r}}_n^{4n}$ is the (relative) trace function from $\text{GF}{2}^{4n}$ to $\text{GF}{2}^n$), we prove that 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 determined for any value in the boomerang spectrum.