## 1 Introduction  
For the effective implementation of digital signal processing (DSP) algorithms, especially digital filters (DF), number-theoretic methods based on prime numbers [1; 2; 3] are of great importance. Many of these methods allow parallel computing, thus the research on theory and application of numerical systems with parallel structure is of particular interest in the field of DSP, image processing systems, cryptographic systems, quantum automated machines, neural computers systems, massive concurrency of operations, cloud computing, etc. [4; 5; 6; 7; 8; 9]. Such systems are best suited for parallel computing. One of the most fruitful research areas here is the algebra of finite field, which provides an impressive level of internal parallelism [10] to the DSP systems.  
Recent studies in designing DSP computing devices based on the finite field algebra (FFA) have shown that FFA has a superior potential for improvement of performance and reliability of numerical information processing being compared with the traditional positional numeral system (PNS) [11].  
Since the FFA is an integer numeral system, it is possible to represent the processed data in DSP devices with arbitrary accuracy. From [11; 12], it is known that the FFA advantages appear most clearly when tabular schemes are in use. Therefore, as the integral technology improves (e.g., the production of storage devices with high information density) along with the technical basis of the tabular computational method, efficiency of using FFA codes is steadily increasing.  
Most popular designs of computing devices operating in FFA codes are focused on the implementation of computational processes of the same type despite their different specialization. These processes are the sequences of arithmetic operations (addition and multiplication) with integer numbers. This determines interest in researching the FFA implementation in highly efficient digital filtering algorithms. The following advantages of FFA can be outlined from [11; 12; 13]:  
1. Independence of formation of numbers' bits. Whereby each bit carries information about the entire original number instead of the intermediate number resulting from the formation of lower-order bits. This implies the possibility of numbers' bits independent parallel processing.  
2. The low bitness of residues representing a number. It results from a small number of possible code combinations. It allows using tabular arithmetic where the typical arithmetic operations are transformed into operations performed by simple selection of the result of calculations from the table (memory device).  
3. The FFA has natural corrective abilities. The FFA codes allow efficient detection and correction of errors while transmitting signals and performing arithmetic operations [14].  
Considering the abovementioned issues, we can conclude that it is advisable to use the FFA for the synthesis of DF with the required quality indicators. The FFA advantages compared with PNS allow providing the required frequency and accuracy characteristics of filters and digital signal processing in real-time [13]. Modern means of digital signal processing (for example, digital receivers) have strict requirements for the quality of signal processing. The analysis carried out in [12; 13] showed that the use of FFA allows one to achieve the required indicators of signal processing quality. Let us give some preliminaries first.