In recent years, stream cipher systems that have been traditionally designed using linear feedback shift register have been almost entirely compromised by algebraic attack methods. Thus, identifying a method to establish concepts for new-generation ciphers, prevent existing security problems, and design new stream cipher systems that consider both security and performance has become a crucial concern in the field of cryptography. In 2004, the European Union initiated the eSTREAM project to emulate the Advanced Encryption Standard used in the United States. The project consisted of 48 participating stream cipher candidates. Through open selection, review, and runoff voting, the results were announced in May 2008. This research investigated one of the finalists of the eSTREAM competition: The Rabbit stream cipher. Additionally, stream cipher attack methods have been extensively studied in recent years, especially those for distinguishing attacks. Thus, contributions of this article is explored the design concepts of the core algorithms in the new-generation stream cipher systems for determining the corresponding mathematical principles and practical approaches to contribute to the study of stream cipher systems.