This article presents examples of an application of the finite field method for the computation of the characteristic polynomial of the matching arrangement of a graph. Weight functions on edges of a graph with weights from a finite field are divided into proper and improper functions in connection with proper colorings of vertices of the matching polytope of a graph. An improper weight function problem is introduced, a proof of its NP-completeness is presented, and a knapsack-like public key cryptosystem is constructed based on the improper weight function problem.

翻译：本文展示了有限域方法在计算图的匹配配置特征多项式中的应用实例。图中边的权重函数根据权重取自有限域的情况，被划分为适当函数与不当函数，这与图的匹配多面体顶点适当着色问题相关联。本文引入了不当权重函数问题，给出了其NP完全性的证明，并基于不当权重函数问题构建了一个类背包公钥密码系统。