Boomerang uniformity of power permutations and algebraic curves over 𝔽2n

Abstract

Abstract We obtain the Boomerang Connectivity Table of power permutations F(x)=x2m−1 of F2n $F(x)={{x}^{{{2}^{m}}-1}}\text{ }\!\!~\!\!\text{ of }\!\!~\!\!\text{ }{{\mathbb{F}}_{{{2}^{n}}}}$with m ∈ { 3,n−12,n+12,n−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∈F2n∖F2. $F(x)\text{ }\!\!~\!\!\text{ at }\!\!~\!\!\text{ }b\in {{\mathbb{F}}_{{{2}^{n}}}}\setminus {{\mathbb{F}}_{2}}.$Moreover we determine the complete Boomerang distribution spectrum of F(x) using the number of rational points of certain concrete algebraic curves over F2n. ${{\mathbb{F}}_{{{2}^{n}}}}.$We also determine the distribution spectra of Boomerang uniformities explicitly.