Compactly supported functions in cryptography algorithms

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 $a$.