The stream ciphers are an important tool for providing information security in the present communication and computer networks. Due to this reason our paper describes a multicore computer system for design of stream ciphers based on the so - named random feedback shift registers (RFSRs). The interest to this theme is inspired by the following facts. First, the RFSRs are a relatively new type of stream ciphers which demonstrate a significant enhancement of the crypto – resistance in a comparison with the classical stream ciphers. Second, the studding of the features of the RFSRs is in very initial stage. Third, the theory of the RFSRs seems to be very hard, which leads to the necessity RFSRs to be explored mainly by the means of computer models. The paper is organized as follows. First, the basics of the RFSRs are recalled. After that, our multicore computer system for design of stream ciphers based on RFSRs is presented. Finally, the advantages and possible areas of application of the computer system are discussed.