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

- 1 Citations
- 723 Downloads

## 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.

## Keywords

Cyber security Defense-in-depth Game over Information security Layered security Weighted rooted trees Complexity Polynomial time Pseudo-polynomial time## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Aghezzaf, E.H., Magnanti, L.T., Wolsey, A.L.: Optimizing Constrained Subtrees of Trees. Mathematical Programming
**71**(2), 113–126 (1995). Series ACrossRefGoogle Scholar - 2.Agnarsson, G., Greenlaw, R.: Graph Theory: Modeling, Applications, and Algorithms. Pearson Prentice Hall, Upper Saddle River (2007)Google Scholar
- 3.Armstrong, R.C., Mayo, J.R., Siebenlist, F.: Complexity Science Challenges in Cybersecurity. Sandia Report, March 2009Google Scholar
- 4.Chakrabarti, D., Faloutsos, C.: Graph Mining: Laws, Generators, and Algorithms. ACM Computing Surveys 38(1), article 2, 69 pages (2006)Google Scholar
- 5.Coene, S., Filippi, C., Spieksma, F., Stevanato, E.: Balancing Profits and Costs on Trees. Networks
**61**(3), 200–211 (2013)CrossRefGoogle Scholar - 6.2012 Cost of Cyber Crime Study: United States. Ponemon Institute, research report, 29, October 2012Google Scholar
- 7.Dunlavy, D.M., Hendrickson, B., Kolda, T.G.: Mathematical Challenges in Cybersecurity. Sandia Report, February 2009Google Scholar
- 8.Hsieh, S.-Y., Chou, Ting-Yu.: Finding a weight-constrained maximum-density subtree in a tree. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 944–953. Springer, Heidelberg (2005)CrossRefGoogle Scholar
- 9.Johnston, R., La Fever, C.,: Hacker.mil, Marine Corps Red Team (Power- Point Presentation) (2012)Google Scholar
- 10.Lau, H.C., Ngo, T.H., Nguyen, B.N.: Finding a Length- constrained Maximum-sum or Maximum-density Subtree and Its Application to Logistics. Discrete Optimization
**3**(4), 385–391 (2006)CrossRefGoogle Scholar - 11.Pfleeger, S.L.: Useful Cybersecurity Metrics. IT Professional
**11**(3), 38–45 (2009)CrossRefGoogle Scholar - 12.Rue, R., Pfleeger, S.L., Ortiz, D.: A framework for clas- sifying and comparing models of cybersecurity investment to support policy and decision-making. In: Proceedings of the Workshop on the Economics of Information Security, p. 23 (2007)Google Scholar
- 13.Schneider, F.B.: Blueprint for a Science of Cybersecurity. The Next Wave
**19**(2), 47–57 (2012)Google Scholar - 14.Shiva, S., Roy, S., Dasgupta, D.: Game theory for cyber security. In: Proceedings of the ACM 6th Annual Cyber Security and Information Intelligence Research Workshop, 34, April 21–23, 2010Google Scholar
- 15.Sparrows, P.: Cyber Crime Statistics. hackmageddon.com, October 16, 2013
- 16.Hsin-Hao, S., Lung, C.H., Tang, C.Y.: An Improved Algorithm for Finding a Length-constrained Maximum-density Subtree in a Tree. Information Processing Letters
**109**(2), 161–164 (2008)CrossRefGoogle Scholar - 17.Agnarsson, G., Greenlaw, R., Kantabutra, S.: On cyber attacks and the maximum-weight rooted-subtree problem. Acta Cybernetica (to appear)Google Scholar