Affiliation:
1. Dept. of Mathematics, Univ. of Paris VIII , 93526 Saint-Denis , France
2. Department of Mathematics and Institute of Applied Mathematics, Middle East Technical University , Ankara , Turkey
Abstract
Abstract
We obtain the Boomerang Connectivity Table of power permutations
F
(
x
)
=
x
2
m
−
1
of
F
2
n
$F(x)={{x}^{{{2}^{m}}-1}}\text{ }\!\!~\!\!\text{ of }\!\!~\!\!\text{ }{{\mathbb{F}}_{{{2}^{n}}}}$
with m ∈
{
3
,
n
−
1
2
,
n
+
1
2
,
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
∈
F
2
n
∖
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.