We embark into theoretical approaches for the investigation of intrusion detection schemes. Our main motivation is to provide rigorous security requirements for intrusion detection systems that can be used by designers of such systems. Our model captures and generalizes well-known methodologies in the intrusion detection area, such as anomaly-based and signature-based intrusion detection, and formulates security requirements based on both well-known complexity-theoretic notions and well-known notions in cryptography (such as computational indistinguishability). Under our model, we present two efficient paradigms for intrusion detection systems, one based on nearest neighbor search algorithms, and one based on both the latter and clustering algorithms. Under formally specified assumptions on the representation of network traffic, we can prove that our two systems satisfy our main security requirement for an intrusion detection system. In both cases, while the potential truth of the assumption rests on heuristic properties of the representation of network traffic (which is hard to avoid due to the unpredictable nature of external attacks to a network), the proof that the systems satisfy desirable detection properties is rigorous and of probabilistic and algorithmic nature. Additionally, our framework raises open questions on intrusion detection systems that can be rigorously studied. As an example, we study the problem of arbitrarily and efficiently extending the detection window of any intrusion detection system, which allows the latter to catch attack sequences interleaved with normal traffic packet sequences. We use combinatoric tools such as time and space-efficient covering set systems to present provably correct solutions to this problem.
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