一类完全非线性函数的原像分布值方程的解

The Solutions of the Equation of Preimage Distribution for a Class of Perfect Nonlinear Functions

求解完全非线性函数的原像分布,是决定和分析完全非线性函数以及构造相应线性码的公开问题和重要课题之一.研究了完全非线性函数的原像分布所满足的基本方程的当m=7时的情形,求出了当n=7^r、n=2^s.7^r以及n=3^s.7^r时方程的所有整数解.

Solving the preimage distribution of a perfect nonlinear function,is very useful for both the existence of perfect nonlinear functions and constructing some linear codes based on perfect nonlinear functions.This is a well-known open problem.In this paper,we discuss the equation of the preimage distribution of a perfect nonlinear function when m=7.We obtain its all integer solutions when n=7^r、n=2^s·7^r and n=3^s·7^r.