幂函数型完全非线性函数原像分布的特征

摘要

完全非线性函数是特征为奇数的有限域上抗差分密码攻击最优的函数,目前已有的六类完全非线性函数都是2-1的.当Ⅱ(x)为Fqm上的Dembowski-Ostrom函数或者Coulter-Matthews函数时,从Fpm到Fq的完全非线性函数tr(aⅡ(x))的原像分布恰有两种取值,其中一种取值对应Fqm所有平方剩余元,另一种取值对应Fqm所有非平方剩余元.该结论在文中得到了证明.%Perfect nonlinear function is the optimal function on finite fields with odd character that can resist differential cryptanalysis. All the six classes of the already known perfect functions currently are 2-1. This study proved that when is Dembowski-Ostrom function or Coulter-Matthews function on , the preimage distribution of the perfect nonlinear function has just two kinds of values, one corresponds to all the elements of quadratic residual on , and the other kind corresponds to all the non quadratic residual on .