The Arithmetical Properties Of Orbits And Applications In Cryptography

Orbit is an important research branch and plays an important role in finite group and cryptology. On the one hand, a specical kind of orbit)the conjugacy classes have been studied in the finite groups. Using the arithmetical condition of the conjugacy classes, we investigate the structure of the normal subgroup with the two longest G-conjugacy classes;On the other hand, as a application of the arithmetical condition of the orbits, we count the balanced rotation symmetirc Boolean functions which have two special linear structures.Several new results are obtained. The main results of this paper are the following:1. In the theory of finite groups, the conjugacy classes as a special orbit play an important role in the research of the structure of finite groups. In chapter 3 of this paper,mainly from the arithmetical condition of the two longest G-conjugacy classes of N, the structure of N has been studied.2. As a application of the arithmetical condition of the orbits, in chapter 4 of this paper, based on the calculation of rotation symmetric orbits and self-conjugate orbits and combined with the properties of the special linear structures, we count the di?erent variable balanced rotation symmetric Boolean functions which have two special linear structures.