## 1 Introduction  
In this paper, we continue the research described in the work [1]. In the presented work, a fast method for determining the sign of a number was developed based on the Akushsky Core Function (ACF). The developed method was based on the calculation of the approximate rank of a number. However, after exploring the basics of ACF, it was found that additional techniques can be applied. The calculation of the approximate rank really makes it possible to increase the efficiency of calculating operations in the Residue Number System (RNS). This is confirmed by several scientific papers that have explored the different applicabilities of the rank of the RNS number. For example, in the article [2], RNS properties are used when developing a sign detection function for homomorphic encryption. RNS is used here to speed up the arithmetic of homomorphic encryption, which also allows the use of RNS techniques such as the rank of a number and its positional characteristic. In another article [3], the polynomial form of RNS (PRNS) is used to improve the reliability of cloud storage. Using PRNS as well as the entropy paradigm, the authors present a method to increase reliability by increasing the fault tolerance of the system due to the self-correcting properties of RNS. The result, according to the authors, allows you to correct errors as well as hardware and software failures. In addition, the presented method allows you to deal with integrity violations and the consequences of attacks and intrusions into the system. In the article [4], research is being undertaken related to the rank of the RNS number. The work presentsproof that a more efficient calculation of the rank is possible based on approximate methods. In addition, the authors demonstrate that the rank value can be obtained with the necessary accuracy. Thus, this study is focused on a more detailed consideration of ACF within RNS, as well as its mathematical features and the possibility of applying them in practice. We are continuing research aimed at improving the reliability of Fog Computing (FC) [5], which will allow more widespread use of FC technology. Fog computing (FC) is a way of organizing distributed computing based on interconnected low-power (edge) devices. Typical examples of FC are Smart City (SM) [6] and Internet of Things (IoT) [7] networks, where several low-power devices consist of various sensors (for example, humidity, temperature, light level, etc. ), as well as traffic lights or, for example, mobile devices and on-board computers of cars. Thus, these low-power devices in large numbers create a computing network capable of processing a certain data stream, for example, creating the computing power necessary for the self-service needs of the IoT network SM. Given the limited computing power of a single device, certain requirements are imposed on the network architecture. Various methods of orchestration and load balancing allowed for increased efficiency in the architecture of FC networks. For example, in [8; 9; 10], the authors study orchestration methods for FC. There are also studies on load balancing in FC; for this, you can read the following works [11; 12; 13; 14; 15]. This made it possible to increase the efficiency of deploying the data transmission infrastructure. Reliability is affected by optimal load distribution between nodes and data transmission channels. However, in this case, the reliability is increased by manipulating the 2-7 levels of the OSI model. However, the first level, the physical level, is not considered by such methods. At the physical level, reliability is determined by the fault tolerance of both the equipment and the information presentation system. RNS here demonstrates its effectiveness due to the properties of self-correction, which allow neglecting errors and anomalies of the physical layer up to a certain threshold. In addition, the case of FC-RNS demonstrates efficiency due to its natural parallelism.