The Complexity of Cyber Attacks in a New Layered-Security Model and the Maximum-Weight, Rooted-Subtree Problem

Abstract

This paper makes three contributions to cyber-security research. First, we define a model for cyber-security systems and the concept of a cyber-security attack within the model’s framework. The model highlights the importance of game-over components—critical system components which if acquired will give an adversary the ability to defeat a system completely. The model is based on systems that use defense-in-depth/layered-security approaches, as many systems do. In the model we define the concept of penetration cost}, which is the cost that must be paid in order to break into the next layer of security. Second, we define natural decision and optimization problems based on cyber-security attacks in terms of doubly weighted trees, and analyze their complexity. More precisely, given a tree T rooted at a vertex r, a penetrating cost edge function c on T, a target-acquisition vertex function p on T, the attacker’s budget and the game-over threshold B ,G∈Q ⁺ respectively, we consider the problem of determining the existence of a rooted subtree T’ of T within the attacker’s budget (that is, the sum of the costs of the edges in T’ is less than or equal to B) with total acquisition value more than the game-over threshold (that is, the sum of the target values of the nodes in T’ is greater than or equal to G). We prove that the general version of this problem is intractable. We also analyze the complexity of three restricted versions of the problems, where the penetration cost is the constant function, integer-valued, and rational-valued among a given fixed number of distinct values. Using recursion and dynamic-programming techniques, we show that for constant penetration costs an optimal cyber-attack strategy can be found in polynomial time, and for integer-valued and rational-valued penetration costs optimal cyber-attack strategies can be found in pseudo-polynomial time. Third, we provide a list of open problems relating to the architectural design of cyber-security systems and to the model.