Algorithms of information authentication for the automatic control systems on the basis of structures in finite vector spaces

New algebraic structures differing from the existing ones in new kinds of operations over finite sets of $m$-dimensional vectors $(m\ge2)$ with discrete components were developed to construct information authentication algorithms for the automatic control systems. The operation of vector multiplication including multiplication and summation of the coordinates of the multiplied vectors modulo simple number $p$ was defined. Depending on the structure of module $p$, dimensionality $m$, and value of the “extension” coefficients defined in terms of the rules of multiplication of the base vectors, the vector spaces were shown to include high-order finite groups or be extended fields that can be efficiently used to construct effective algorithms of electronic digital signature.