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Boomerang uniformity of power permutations and algebraic curves over 𝔽2n

Boomerang uniformity of power permutations and algebraic curves over 𝔽2n AbstractWe 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Geometry de Gruyter

Boomerang uniformity of power permutations and algebraic curves over 𝔽2n

Mesnager, Sihem; Özbudak, Ferruh
Advances in Geometry , Volume 23 (1): 28 – Jan 1, 2023
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Publisher
de Gruyter
Copyright
© 2023 Walter de Gruyter GmbH, Berlin/Boston
eISSN
1615-715X
DOI
10.1515/advgeom-2022-0026
Publisher site
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Abstract

AbstractWe 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.

Journal

Advances in Geometry – de Gruyter

Published: Jan 1, 2023

Keywords: Boomerang uniformity; differential uniformity; algebraic curve; algebraic function field; 11T71; 94A60; 11G20

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