有限域上随机变量联合分布及二阶矩的分解与应用

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

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Ｘ平方统计量与该随机向量坐标函数的非零线性组合的Ｘ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与其差分分布之间的内在联系。

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.