Geodesic
Compactly supported functions in cryptography algorithms
Prikladnaâ diskretnaâ matematika, no. 2 (2017), pp. 73-83.

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

In the paper, we propose a new symmetric block cipher based on the orthogonal finite functions (OFF) in the Sobolev's space. To encrypt a plaintext $a=a_1a_2\dots a_n$, we first convert $a$ to a polynomial $a(x)=a_1+a_2x+\dots+a_nx^{n-1}$, then approximate $a(x)$ by a linear combination $F(x)=\sum_{i=1}^nr_if_i(x)$, where $f=(f_1,f_2,\dots,f_n)$ is an OFF-basis not having the orthogonality property, and finally compute the ciphertext $b=b_1b_2\dots b_n$, where $b_i=F(k_i)$ for some values $k_i$ of $x$, $i=1,2,\dots,n$. The numbers $k_1,\dots,k_n$ and some parameters of functions in $f$ form the key of the cipher. To decrypt the ciphertext $b$, we first, given $b_1,\dots,b_n$ and key parameters in $f$, compute the approximation coefficients $r_1,\dots,r_n$, next, given $k_1,\dots,k_n$, compute $x'_1,\dots,x'_n$ such that $a(x'_i)=r_i$ for $i=1,2,\dots,n$, then construct $a(x)$ by the Lagrange method, and finally convert $a(x)$ to $a$.
Keywords: OFF, finite functions, cryptography, encryption. 
@article{PDM_2017_2_a5,
     author = {A. V. Shchurenko and V. L. Leontiev},
     title = {Compactly supported functions in cryptography algorithms},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {73--83},
     publisher = {mathdoc},
     number = {2},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDM_2017_2_a5/}
}
TY  - JOUR
AU  - A. V. Shchurenko
AU  - V. L. Leontiev
TI  - Compactly supported functions in cryptography algorithms
JO  - Prikladnaâ diskretnaâ matematika
PY  - 2017
SP  - 73
EP  - 83
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PDM_2017_2_a5/
LA  - ru
ID  - PDM_2017_2_a5
ER  - 
%0 Journal Article
%A A. V. Shchurenko
%A V. L. Leontiev
%T Compactly supported functions in cryptography algorithms
%J Prikladnaâ diskretnaâ matematika
%D 2017
%P 73-83
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PDM_2017_2_a5/
%G ru
%F PDM_2017_2_a5
A. V. Shchurenko; V. L. Leontiev. Compactly supported functions in cryptography algorithms. Prikladnaâ diskretnaâ matematika, no. 2 (2017), pp. 73-83. http://geodesic.mathdoc.fr/item/PDM_2017_2_a5/

[1] Leont'ev V. L., Orthogonal finite functions and numerical methods, UlSU Publ., Ul'yanovsk, 2003, 178 pp. (in Russian) | MR

[2] Leont'ev V. L., Lukashenets N. Ch., “Grid bases of orthogonal compactly supported functions”, Zh. Vychisl. Mat. Mat. Fiz., 39:7 (1999), 1158–1168 (in Russian) | MR | Zbl

[3] Leont'ev V. L., “Orthogonal splines and variational-grid method”, Matem. Mod., 14:3 (2002), 117–127 (in Russian) | MR | Zbl

[4] Lukomskiy D. S., Lukomskiy S. F., “Splash bases and cryptography”, Matematika. Mekhanika, 2011, no. 13, 55–58 (in Russian)

[5] Levina A. B., Spline-Wavelets and some their Applications, PhD Thesis, SPb., 2009, 214 pp. (in Russian)