中文摘要：

利用代数数论的有关知识与理论,研究了从3l阶交换群到3阶交换群上完全非线性函数的原像分布特征方程,通过讨论其等价方程x~2＋xy＋y~2=l的整数解问题,给出了该类完全非线性函数存在的必要条件及其原像分布特征的计数.进一步给出了求该类完全非线性函数所有可能原像分布特征的一个算法.

英文摘要：

By using the algebraic number theory,the preimage distributions of perfect nonlinear functions are discussed,the functions map an Abel group of order 3l to an Abel group of order 3.The integral roots of the equivalent equation x~2＋ xy ＋ y~2= l are investigated,and a necessary condition for the existence of this kind of functions and the enumeration of preimage distributions are given. Further,an algorithm which can be used to give all possible preimage distributions of this kind of functions is presented.