Geodesic
On the relationship between nonlinear and differential properties of vectorial Boolean functions
Prikladnaya Diskretnaya Matematika. Supplement, no. 12 (2019), pp. 60-62

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

The relations between the linear approximation table (LAT) and the differences distribution table (DDT) of the vectorial Boolean functions are investigated. Let $F$ be a function from $\mathbb{F}_2^n$ into $\mathbb{F}_2^n$. DDT of $F$ is a $2^n\times 2^n$ table defined by DDT$(a, b) = |\{x\in\mathbb{F}_2^n | F(x) \oplus F(x\oplus a) = b \}|$ for each $a,b\in \mathbb{F}_2^n$. LAT of $F$ is a $2^n \times 2^n$ table, in the cell $(v, u)$ of which the squared Walsh — Hadamard coefficient is stored. It is proved that the presence of coinciding rows in DDT and LAT is an invariant under affine equivalence as well as under EA-equivalence for normalized DDT and LAT. It is hypothesized that if all rows in the LAT (DDT) of a vectorial Boolean function $F$ are pairwise different, then all rows in its DDT (LAT) are also pairwise different. This hypothesis is checked for functions in a small number of variables and for known APN functions in not more than 10 variables.
Keywords: APN function, AB function, differential uniformity, nonlinearity. 
A. V. Miloserdov. On the relationship between nonlinear and differential properties of vectorial Boolean functions. Prikladnaya Diskretnaya Matematika. Supplement, no. 12 (2019), pp. 60-62. http://geodesic.mathdoc.fr/item/PDMA_2019_12_a17/
@article{PDMA_2019_12_a17,
     author = {A. V. Miloserdov},
     title = {On the relationship between nonlinear and differential properties of vectorial {Boolean} functions},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {60--62},
     year = {2019},
     number = {12},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2019_12_a17/}
}
TY  - JOUR
AU  - A. V. Miloserdov
TI  - On the relationship between nonlinear and differential properties of vectorial Boolean functions
JO  - Prikladnaya Diskretnaya Matematika. Supplement
PY  - 2019
SP  - 60
EP  - 62
IS  - 12
UR  - http://geodesic.mathdoc.fr/item/PDMA_2019_12_a17/
LA  - ru
ID  - PDMA_2019_12_a17
ER  - 
%0 Journal Article
%A A. V. Miloserdov
%T On the relationship between nonlinear and differential properties of vectorial Boolean functions
%J Prikladnaya Diskretnaya Matematika. Supplement
%D 2019
%P 60-62
%N 12
%U http://geodesic.mathdoc.fr/item/PDMA_2019_12_a17/
%G ru
%F PDMA_2019_12_a17

[1] Biham E., Shamir A., “Differential cryptanalysis of DES-like cryptosystems”, J. Cryptology, 4:1 (1991), 3–72 | DOI | MR | Zbl

[2] Matsui M., Yamagishi A., “A new method for known plaintext attack of FEAL cipher”, EUROCRYPT'1992, LNCS, 658, 1992, 81–91

[3] Carlet C., “Vectorial Boolean functions for cryptography”, Boolean Models and Methods in Mathematics, Computer Science, and Engineering, eds. Y. Crama, P. Hammer, Cambridge University Press, Cambridge, 2010, 398–470 | DOI | MR