关于代数次数大于2的旋转对称bent函数的研究这两年才开始起步，还有许多关键的问题需要解决。本项目的主要工作是：1）到目前为止，据我们所知已经构造出了三类非常特殊的旋转对称bent函数，本项目将继续研究任意可能代数次数、任意偶数变元的旋转对称bent函数的构造方法以及有限域上的幂等bent函数的构造方法；2）研究奇数变元的旋转对称semi-bent函数的构造方法，然后通过修改前面所构造的旋转对称bent函数以及旋转对称semi-bent函数的支撑集来构造非线性度比semi-bent函数的非线性度还高的旋转对称布尔函数，并且满足1阶弹性等密码学性质；3）现在已知的具有最优代数免疫度的旋转对称布尔函数的非线性度都太低了，本项目将通过修改前面这几类函数的支撑集来构造具有最优代数免疫度的旋转对称布尔函数，而且保证非线性度能够充分接近bent函数的非线性度。

The study of nonquadratic rotation symmetric bent functions arises just in the past two years, but there are still many key problems need to be solved. The main work of this project is as follows : 1) So far, three classes of very particular rotation symmetric bent functions were constructed. This project will study how to construct even-variable rotation symmetric bent functions with arbitrary possible algebraic degrees and how to construct idempotent bent functions defined on the finite field. 2) This project will study how to construct odd-variable rotation symmetric semi-bent functions, and then by modifying the outputs of the rotation symmetric bent functions as well as the rotation symmetric semi-bent functions we will construct new rotation symmetric Boolean functions whose nonlinearities are higher than the nonlinearity of semi-bent functions, satisfying the first-order resilience and other cryptographic properties; 3) The nonlinearities of the known rotation symmetric Boolean functions with optimal algebraic immunity are too low. This project will study how to modify the supports of the functions we get above to construct new rotation symmetric Boolean functions with optimal algebraic immunity, whose nonlinearities are very close to the nonlinearity of bent functions.