有限域上随机变量联合分布及二阶矩的分解与应用

本文给出了有限域上随机变量联合概率和二阶矩的分解公式，给出了有限域上随机变量相互独立的谱刻划，应用上述结果，建立了在进行频次分析时，对有限域上随机向量构造的Χ平方统计量与该随机向量坐标函数的非零线性组合的Χ平方统计量之间的内在联系，给出了有限域上相关免疫函数谱特征的新证明，建立了有限域上多输出函数的差分分布与其广义Chrestenson循环谱之间的内在联系，建立了多输出函数的平衡性其差分分布之间的内在联系。

DECOMPOSITIONS OF THE JOINT DISTRIBUTIONS AND OF THE SECOND MOMENTS FOR RANDOM VARIABLES OVER FINITE FIELDS AND THEIR APPLICATIONS

In this paper, the authors prove the decomposition formulas for the joint dis- tributions of sets of random variables over finite fields and for their second moments, and gives a spectral characterization for independence of random variables over finite fields. As applications, the authors established an intrinsic relation between the x square statistics of a random vector on finite fields and the x square statistics of the nonzero linear combina- tions of coordinates of this random vector constructed when one performs a frequency test, and gives a new proof for the spectral characterization of correlation-immunity over finite fields, establish the links between differential distribution and Chrestenson cyclic spectral and the link between balance and differential distribution for multiple output functions on finite fields.