The problem of designing a secure system of protection against unauthorized access is considered. In particular, this article considers hierarchical data protection systems with cryptographic key distribution, namely, the problem of organizing access to file storages is considered. Although cryptographic key distribution can ensure the security of information from users who do not have access to it, the hierarchical access control system was not originally designed to solve the problem of protecting information from the dishonest actions of the user himself. Thus, the overall objective of the study is to prevent unauthorized exchange of information between users of different levels of a hierarchical system of protection against unauthorized access with cryptographic key distribution. To achieve the stated goal, the authors previously proposed to use the problems of Diophantine analysis, in particular the knapsack problem. Previously, the authors formulated the properties of the knapsack vector, applicable for improving the hierarchical system of protection against unauthorized access. In this article, the authors present the conditions for the injectivity of knapsack vectors. A comparative analysis of these conditions with the already established injectivity conditions is carried out. The analysis shows the need to formulate such conditions and the applicability of knapsack vectors that satisfy them for improving the hierarchical model of protection against unauthorized access. Based on the specified conditions, this article develops a recursive algorithm for constructing an injective multiplicative knapsack vector. The authors then analyze the possibility of its application for modeling a hierarchical mandatory model of information protection from unauthorized access. The analysis shows how already known algorithms for constructing knapsack vectors can be used as part of the developed algorithm. The authors also show where exactly in the developed system it is necessary to apply this algorithm to implement the properties required for hierarchical systems of protection against unauthorized access
