Boomerang uniformity of power permutations and algebraic curves over ๐ฝ2 n
Sihem Mesnager, Ferruh รzbudak ยท Advances in Geometry ยท 2023
Abstract We obtain the Boomerang Connectivity Table of power permutations F ( x ) = x 2 m โ 1 of F 2 n $F(x)={{x}^{{{2}^{m}}-1}}\text{ }\!\!~\!\!\text{ of }\!\!~\!\!\text{ }{{\mathbb{F}}_{{{2}^{n}}}}$ with m โ { 3 , n โ 1 2 , n + 1 2 , n โ 2 } . $\left\{ 3,\frac{n-1}{2},\frac{n+1}{2},n-2 \right\}.$ In particular, we obtain the Boomerang uniformity and the Boomerang uniformity set of F ( x ) at b โ F 2 n โ F 2 . $F(x)\text{ }\!\!~\!\!\text{ at }\!\!~\!\!\text{ }b\in {{\mathbb{F}}_{{{2}^{n}}}}\setminus {{\mathbb{F}}_{2}}.$ Moreover we determine the complete Boomerang distribution spectrum of F(x) using the number of rational points of certain concrete algebraic curves over F 2 n . ${{\mathbb{F}}_{{{2}^{n}}}}.$ We also determine the distribution spectra of Boomerang uniformities explicitly.