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完全非线性(perfect nonlinear,PN)函数在密码学和代数组合等领域中具有重要的应用价值。从已有的两个基本完全非线性函数出发,通过确定一类有限域上线性化多项式方程的解,构造了奇数特征有限域上一类2次完全非线性函数,证明了该函数不等价于形式为x2的完全非线性函数,并通过一个实例分析了它与完全非线性函数xps+1的等价性问题。
The perfect nonlinear (PN) function has important application value in the fields of cryptography and algebraic combination. Starting from the existing two completely complete nonlinear functions, a class of complete nonlinear functions over a finite number of finite fields is constructed by determining the solution of a linearized polynomial equation over a finite field. It is proved that the function is not equivalent In the form of a complete non-linear function x2, and an example of its complete nonlinear function xps + 1 equivalence problem.