Geodesic
A note on the properties of associated Boolean functions of quadratic APN functions
Prikladnaâ diskretnaâ matematika, no. 1 (2020), pp. 16-21.

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $F$ be a quadratic APN function in $n$ variables. The associated Boolean function $\gamma_F$ in $2n$ variables ($\gamma_F(a,b)=1$ if $a\neq\mathbf{0}$ and equation $F(x)+F(x+a)=b$ has solutions) has the form $\gamma_F(a,b) = \Phi_F(a) \cdot b + \varphi_F(a) + 1$ for appropriate functions $\Phi_F:\mathbb{F}_2^n\to \mathbb{F}_2^n$ and $\varphi_F:\mathbb{F}_2^n\to \mathbb{F}_2$. We summarize the known results and prove new ones regarding properties of $\Phi_F$ and $\varphi_F$. For instance, we prove that degree of $\Phi_F$ is either $n$ or less or equal to $n-2$. Based on computation experiments, we formulate a conjecture that degree of any component function of $\Phi_F$ is $n-2$. We show that this conjecture is based on two other conjectures of independent interest.
Keywords: a quadratic APN function, the associated Boolean function, degree of a function. 
@article{PDM_2020_1_a1,
     author = {A. A. Gorodilova},
     title = {A note on the properties of associated {Boolean} functions of quadratic {APN} functions},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {16--21},
     publisher = {mathdoc},
     number = {1},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PDM_2020_1_a1/}
}
TY  - JOUR
AU  - A. A. Gorodilova
TI  - A note on the properties of associated Boolean functions of quadratic APN functions
JO  - Prikladnaâ diskretnaâ matematika
PY  - 2020
SP  - 16
EP  - 21
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PDM_2020_1_a1/
LA  - en
ID  - PDM_2020_1_a1
ER  - 
%0 Journal Article
%A A. A. Gorodilova
%T A note on the properties of associated Boolean functions of quadratic APN functions
%J Prikladnaâ diskretnaâ matematika
%D 2020
%P 16-21
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PDM_2020_1_a1/
%G en
%F PDM_2020_1_a1
A. A. Gorodilova. A note on the properties of associated Boolean functions of quadratic APN functions. Prikladnaâ diskretnaâ matematika, no. 1 (2020), pp. 16-21. http://geodesic.mathdoc.fr/item/PDM_2020_1_a1/

[1] Nyberg K., “Differentially uniform mappings for cryptography.”, Advances in Cryptography, EUROCRYPT'93, LNCS, 765, 1994, 55–64 | MR | Zbl

[2] Budaghyan L., Construction and Analysis of Cryptographic Functions, Springer International Publishing, 2014, 168 pp. | MR | Zbl

[3] Pott A., “Almost perfect and planar functions”, Designs, Codes and Cryptography, 78 (2016), 141–195 | DOI | MR | Zbl

[4] Glukhov M. M., “On the approximation of discrete functions by linear functions”, Matematicheskie Voprosy Kriptografii, 7:4 (2016), 29–50 (in Russian) | DOI | MR

[5] Tuzhilin M. E., “APN-functions”, Prikladnaya Diskretnaya Matematika, 2009, no. 3(5), 14–20 (in Russian) | DOI | MR

[6] Carlet C., Charpin P., Zinoviev V., “Codes, bent functions and permutations suitable for DES-like cryptosystems”, Designs, Codes and Cryptography, 15:2 (1998), 125–156 | DOI | MR | Zbl

[7] Gorodilova A., “On the differential equivalence of APN functions”, Cryptography and Communications, 11:4 (2019), 793–813 | DOI | MR | Zbl

[8] Boura C., Canteaut A., Jean J., Suder V., “Two notions of differential equivalence on Sboxes”, Designs, Codes and Cryptography, 87:2–3 (2019), 185–202 | DOI | MR | Zbl

[9] Gorodilova A., “The linear spectrum of quadratic APN functions”, Prikladnaya Diskretnaya Matematika, 2016, no. 4(34), 5–16 (in Russian) | DOI | MR

[10] Tokareva N., Bent Functions, Results and Applications to Cryptography, Acad. Press. Elsevier, 2015, 230 pp. | MR | Zbl