Geodesic
$\mathrm{S}$-blocks of a~special type with~a~small~number~of~variables
Diskretnyj analiz i issledovanie operacij, Tome 30 (2023) no. 2, pp. 67-80.

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

When constructing block ciphers, it is necessary to use vector Boolean functions with special cryptographic properties as $\mathrm{S}$-blocks for the cipher's resistance to various types of cryptanalysis. In this paper, we investigate the following $\mathrm{S}$-block construction: let $\pi$ be a permutation on $n$ elements, $\pi^i$ $i$-multiple application $\pi,$ and $f$ a Boolean function in $n$ variables. Define a vectorial Boolean function $F_{\pi}\colon\mathbb{Z}_2^n \to \mathbb{Z}_2^n$ as $F_{\pi}(x) = (f(x), f(\pi(x)), \ldots , f(\pi_{n-1}(x))).$ We study cryptographic properties of $F_{\pi}$ such as high nonlinearity, balancedness, and low differential $\delta$-uniformity in dependence on properties of $f$ and $\pi$ for small $n.$ Complete sets of Boolean functions $f$ and vector Boolean functions $F_{\pi}$ in a small number of variables with maximum algebraic immunity are also obtained. Bibliogr. 16.
Keywords: Boolean functions, vectorial Boolean functions, high nonlinearity, high algebraic degree, balancedness, low differential $\delta$-uniformity, high algebraic immunity. 
@article{DA_2023_30_2_a3,
     author = {D. A. Zyubina and N. N. Tokareva},
     title = {$\mathrm{S}$-blocks of a~special type with~a~small~number~of~variables},
     journal = {Diskretnyj analiz i issledovanie operacij},
     pages = {67--80},
     publisher = {mathdoc},
     volume = {30},
     number = {2},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DA_2023_30_2_a3/}
}
TY  - JOUR
AU  - D. A. Zyubina
AU  - N. N. Tokareva
TI  - $\mathrm{S}$-blocks of a~special type with~a~small~number~of~variables
JO  - Diskretnyj analiz i issledovanie operacij
PY  - 2023
SP  - 67
EP  - 80
VL  - 30
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DA_2023_30_2_a3/
LA  - ru
ID  - DA_2023_30_2_a3
ER  - 
%0 Journal Article
%A D. A. Zyubina
%A N. N. Tokareva
%T $\mathrm{S}$-blocks of a~special type with~a~small~number~of~variables
%J Diskretnyj analiz i issledovanie operacij
%D 2023
%P 67-80
%V 30
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DA_2023_30_2_a3/
%G ru
%F DA_2023_30_2_a3
D. A. Zyubina; N. N. Tokareva. $\mathrm{S}$-blocks of a~special type with~a~small~number~of~variables. Diskretnyj analiz i issledovanie operacij, Tome 30 (2023) no. 2, pp. 67-80. http://geodesic.mathdoc.fr/item/DA_2023_30_2_a3/

[1] Matsui M., “Linear cryptanalysis method for DES cipher”, Advances in cryptology — EUROCRYPT'93, Proc. Workshop Theory and Application of Cryptographic Techniques (Lofthus, Norway, May 23–27, 1993), Lect. Notes Comput. Sci., 765, Springer, Heidelberg, 1994, 386–397 | DOI | MR | Zbl

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

[3] Courtois N. T., Meier W., “Algebraic attacks on stream ciphers with linear feedback”, Advances in cryptology — EUROCRYPT 2003, Proc. Int. Conf. Theory and Application of Cryptographic Techniques (Warsaw, Poland, May 4–8, 2003), Lect. Notes Comput. Sci., 2656, Springer, Heidelberg, 2003, 345–359 | DOI | MR | Zbl

[4] Cusick T. W., Stanica P., Cryptographic Boolean functions and applications, Elsevier, San Diego, CA, 2009, 288 pp. | MR | Zbl

[5] Carlet C., “Vectorial Boolean functions for cryptography”, Boolean models and methods in mathematics, computer science, and engineering, Camb. Univ. Press, Cambridge, 2010, 398–470 | DOI | MR

[6] Tokareva N. N., Bent functions: Results and applications to cryptography, Elsevier, Amsterdam, 2015, 221 pp. | MR

[7] A. A. Gorodilova, “From cryptanalysis to cryptographic property of a Boolean function”, Prikl. Diskretn. Mat., 2016, no. 3, 16–44 (Russian) | MR | Zbl

[8] I. A. Pankratova, Boolean Functions in Cryptography, Izd. Dom Tomsk. Gos. Univ., Tomsk, 2014 (Russian)

[9] N. N. Tokareva, A. A. Gorodilova, S. V. Agievich, et al., “Mathematical methods in solutions of the problems from the Third International Students' Olympiad in Cryptography”, Prikl. Diskretn. Mat., 2018, no. 40, 34–58 | MR | Zbl

[10] Dillon J. F., Elementary Hadamard difference sets, PhD thesis, Univ. Maryland, College Park, 1974 | MR | Zbl

[11] Alsalami Y., “Constructions with high algebraic degree of differentially 4-uniform $(n,n-1)$-functions and differentially 8-uniform $(n,n-2)$-functions”, Cryptogr. Commun., 10:4 (2018), 611–628 | DOI | MR | Zbl

[12] Browning K. A., Dillon J. F., McQuistan M. T., Wolfe A. J., “An APN permutation in dimension six”, Finite fields: Theory and applications, Proc. 9th Int. Conf. Finite Fields and Applications (Dublin, Ireland, July 13–17, 2009), Contemp. Math., 518, AMS, Providence, 2010, 33–42 | DOI | MR | Zbl

[13] Carlet C., “Boolean functions for cryptography and error-correcting codes”, Boolean models and methods in mathematics, computer science, and engineering, Camb. Univ. Press, Cambridge, 2010, 257–397 | DOI | MR | Zbl

[14] Nyberg K., “Differentially uniform mappings for cryptography”, Advances in cryptology — EUROCRYPT'93, Proc. Workshop Theory and Application of Cryptographic Techniques (Lofthus, Norway, May 23–27, 1993), Lect. Notes Comput. Sci., 765, Springer, Heidelberg, 1994, 55–64 | DOI | MR | Zbl

[15] Carlet C., “On the algebraic immunities and higher order nonlinearities of vectorial Boolean functions”, Enhancing cryptographic primitives with techniques from error correcting codes, Proc. NATO Adv. Res. Workshop (Veliko Tarnovo, Bulgaria, Oct. 6–9, 2008), NATO Sci. Peace Secur. Ser. D: Inf. Commun. Secur., 23, IOS Press, Amsterdam, 2009, 104–116 | MR

[16] D. P. Pokrasenko, “On the maximal component algebraic immunity of vectorial Boolean functions”, J. Appl. Ind. Math., 10:2 (2016), 257–263 (Russian) | DOI | MR | Zbl