scholarly journals On the dual of the dual hyperoval from APN function f(x)=x3+Tr(x9)

2012 ◽  
Vol 18 (1) ◽  
pp. 210-221 ◽  
Author(s):  
Hiroaki Taniguchi
Keyword(s):  
Apn Function ◽  
Dual Hyperoval

scholarly journals A new construction of the d-dimensional Buratti–Del Fra dual hyperoval

2012 ◽  
Vol 33 (6) ◽  
pp. 1030-1042 ◽  
Author(s):  
Hiroaki Taniguchi ◽  
Satoshi Yoshiara
Keyword(s):  
New Construction ◽  
Dual Hyperoval

scholarly journals Revisiting some results on APN and algebraic immune functions

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Claude Carlet
Keyword(s):  
Boolean Functions ◽  
The Other ◽  
Immune Functions ◽  
Apn Functions ◽  
Power Functions ◽  
Differential Uniformity ◽  
Open Problems ◽  
Apn Function ◽  
Almost Perfect Nonlinear

<p style='text-indent:20px;'>We push a little further the study of two recent characterizations of almost perfect nonlinear (APN) functions. We state open problems about them, and we revisit in their perspective a well-known result from Dobbertin on APN exponents. This leads us to a new result about APN power functions and more general APN polynomials with coefficients in a subfield <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{F}_{2^k} $\end{document}</tex-math></inline-formula>, which eases the research of such functions. It also allows to construct automatically many differentially uniform functions from them (this avoids calculations for proving their differential uniformity as done in a recent paper, which are tedious and specific to each APN function). In a second part, we give simple proofs of two important results on Boolean functions, one of which deserves to be better known but needed clarification, while the other needed correction.</p>

Doubly transitive dimensional dual hyperovals: universal covers and non-bilinear examples

2019 ◽  
Vol 19 (3) ◽  
pp. 359-379
Author(s):  
Ulrich Dempwolff
Keyword(s):  
Universal Covers ◽  
Dual Hyperoval ◽  
Dimensional Dual Hyperovals

Abstract By [4] a doubly transitive, non-solvable dimensional dual hyperoval D is isomorphic either to the Mathieu dual hyperoval or to a quotient of a Huybrechts dual hyperoval. In order to determine all doubly transitive dimensional dual hyperovals, it remains to classify the solvable ones, and this paper is a contribution to this problem. A doubly transitive, solvable dimensional dual hyperoval D of rank n is defined over 𝔽2 and has an automorphism of the form ES, where E is elementary abelian of order 2n and S ≤ Γ L(1, 2n); see Yoshiara [12]. The known examples D are bilinear. In [1] the bilinear, doubly transitive, solvable dimensional dual hyperovals D of rank n with GL(1, 2n) ≤ S are classified. Here we present two new classes of non-bilinear, doubly transitive dimensional dual hyperovals. We also consider universal covers of doubly transitive dimensional dual hyperovals, since they are again doubly transitive dimensional dual hyperovals by [2, Cor. 1.3]. We determine the universal covers of the presently known doubly transitive dimensional dual hyperovals.

scholarly journals A new APN function which is not equivalent to a power mapping

2006 ◽  
Vol 52 (2) ◽  
pp. 744-747 ◽  
Author(s):  
Y. Edel ◽  
G. Kyureghyan ◽  
A. Pott
Keyword(s):  
Apn Function ◽  
Power Mapping

On the Walsh Spectrum of a New APN Function

2007 ◽  
pp. 92-98 ◽  
Author(s):  
Carl Bracken ◽  
Eimear Byrne ◽  
Nadya Markin ◽  
Gary McGuire
Keyword(s):  
Walsh Spectrum ◽  
Apn Function

The non-solvable doubly transitive dimensional dual hyperovals

2018 ◽  
Vol 18 (1) ◽  
pp. 1-4
Author(s):  
Ulrich Dempwolff
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Elementary Abelian Group ◽  
Dual Hyperoval ◽  
Dimensional Dual Hyperovals

AbstractIn [9] S. Yoshiara determines possible automorphism group of doubly transitive dimensional dual hyperovals. He shows that a doubly transitive dual hyperovalDis either isomorphic to the Mathieu dual hyperoval or the dual hyperoval is defined over 𝔽2, and if the hyperoval has rankn, the automorphism group has the formE⋅S, with an elementary abelian groupEof order 2nandSa subgroup of GL(n,2) acting transitively on the nontrivial elements ofE. Moreover Yoshiara describes the possible candidates forS. In this paper we assume thatSis non-solvable and show that then the dimensional dual hyperoval is a bilinear quotient of a Hyubrechts dual hyperoval.

scholarly journals On the exponents of APN power functions and Sidon sets, sum-free sets, and Dickson polynomials

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Claude Carlet ◽  
Stjepan Picek
Keyword(s):  
Necessary Conditions ◽  
Power Functions ◽  
Apn Function ◽  
Dickson Polynomials ◽  
Sidon Sets ◽  
Dickson Polynomial ◽  
Speed Up ◽  
Reversed Dickson Polynomial ◽  
Characteristic 2 ◽  
Free Sets

<p style='text-indent:20px;'>We derive necessary conditions related to the notions, in additive combinatorics, of Sidon sets and sum-free sets, on those exponents <inline-formula><tex-math id="M1">\begin{document}$ d\in {\mathbb Z}/(2^n-1){\mathbb Z} $\end{document}</tex-math></inline-formula>, which are such that <inline-formula><tex-math id="M2">\begin{document}$ F(x) = x^d $\end{document}</tex-math></inline-formula> is an APN function over <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb F}_{2^n} $\end{document}</tex-math></inline-formula> (which is an important cryptographic property). We study to what extent these new conditions may speed up the search for new APN exponents <inline-formula><tex-math id="M4">\begin{document}$ d $\end{document}</tex-math></inline-formula>. We summarize all the necessary conditions that an exponent must satisfy for having a chance of being an APN, including the new conditions presented in this work. Next, we give results up to <inline-formula><tex-math id="M5">\begin{document}$ n = 48 $\end{document}</tex-math></inline-formula>, providing the number of exponents satisfying all the conditions for a function to be APN.</p><p style='text-indent:20px;'>We also show a new connection between APN exponents and Dickson polynomials: <inline-formula><tex-math id="M6">\begin{document}$ F(x) = x^d $\end{document}</tex-math></inline-formula> is APN if and only if the reciprocal polynomial of the Dickson polynomial of index <inline-formula><tex-math id="M7">\begin{document}$ d $\end{document}</tex-math></inline-formula> is an injective function from <inline-formula><tex-math id="M8">\begin{document}$ \{y\in {\Bbb F}_{2^n}^*; tr_n(y) = 0\} $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M9">\begin{document}$ {\Bbb F}_{2^n}\setminus \{1\} $\end{document}</tex-math></inline-formula>. This also leads to a new and simple connection between Reversed Dickson polynomials and reciprocals of Dickson polynomials in characteristic 2 (which generalizes to every characteristic thanks to a small modification): the squared Reversed Dickson polynomial of some index and the reciprocal of the Dickson polynomial of the same index are equal.</p>

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