Discrete Functions

A conditions for uniqueness reresentation of $p$-logic function into disjunctive product of functions

A. V. Cheremushkin

Academy of Cryptography of Russian Federation

Abstract: Let $f:V_n \rightarrow {\mathbb Z}_p$ be $p$-logic function, $n\ge 2$, and $V_n={\mathbb Z}_p^n$ is considered as a vector space over ${\mathbb Z}_p$. A disjunctive decomposition of $f$ into a product of $p$-logic functions under various linear transformations of arguments is considered. Function $f$ is linearly decomposable into disjunctive product if there exists a linear transformation $A$ of the vector space $V_n$ such that
$$ f(xA)= f_1(x_1,\ldots , x_k) f_2(x_{k+1},\ldots , x_n) $$
for some $k$, $1\le k <n$, and functions $f_1$ and $f_2$. We say that argument $x_n$ of functions $f(x)$ is essential iff $f(x)\neq f(x + e_n)$ for $e_n=(0,\ldots, 0,1)$. The main result is: if all arguments of all functions $f(xA)$ under linear substitutuions $A$ of the vector space $V_n$ are essential, the set $\{a\in V_n: f(a)\neq 0\}$ is not contained in hyperplane of $V_n$, and $f$ is linearly decompsable into the disjunctive product $f_1\cdot \dots \cdot f_m$, where $m$ is maximal, then the direct sum of subspaces $V_n=V^{(1)}+\ldots +V^{(m)}$ is unique and invariant under the stabilizer group of the function $f$ in general linear group.

Keywords: $p$-logic functions, disjunctive product, linear transformation.

UDC: 519.719.325

DOI: 10.17223/2226308X/14/10