The arithmetic of the topologist's sine curve in cryptographic systems dedicated to IoT devices

When observing the modern world, we can see the dynamic development of new technologies, among which a special place, both owing to the potential and the threats is occupiedby the Internet of Things which penetrates almost all areas of our life. It is assumed that the IoT technology makes our life easier, however, it poses many challenges concerning the protection of the security of information transmission and, therefore, our privacy. One of the main goals of the paper is to present a new unconventional arithmetic based onthe transcendental curve dedicated to the cryptographic systems that protect the transmission of short messages. The use of this arithmetic may develop the possibilities of protecting short sequences of data generated by devices with limited computational power. Examples of such devices include the ubiquitously used battery powered sensors, the task of which is to collectand transmit data which very often comprises concise information. Another goal is to present the possibility of using the developed arithmetic in cryptographic algorithms