中文摘要：

为了将高度非线性函数应用于认证码构造理论．讨论了非线性函数的度量，给出了由三类非线性函数构造的Cartesian认证码，并且当编码规则按等概率分布选取时，分别计算了三类认证码的模仿攻击和欺骗攻击成功的概率。结果表明，第二类认证码模仿攻击成功的概率达到了下界。在n取值相同的条件下，第一类认证方案模仿攻击成功的概率比第三类的大。

英文摘要：

In order to construct authentication codes by applying functions with high nonlinearity, we have discussed the measure of the non-linear functions, given the the construction of Cartesian authentication code from three types of non-linear function. Under the assumption that the encoding rules of the transmitter and the receiver are chosen according to a uniform probability distribution, the maximum probabilities of success with respect to the impersonation and substitution attacks are also computed, respectively. The results show that the probability of success of impersonation attack in the second type of authentication codes reached the lower bound. Under n value the same conditions, the probability of success of impersonation attack in the first type of authentication codes is larger than the third.