“Cyber ShockWave…featured a number of former US government officials who played the part of senior members of the NSC. The exercise sought to examine how the NSC would react to a major cyber attack in real time…the source of the attack remained unclear during the event…The mock NSC even discussed potentially nationalizing power companies and service providers if they failed to act in the national interest. Ultimately, in the several hours that the war game lasted, the US was increasingly beset by attack with little knowledge of who perpetrated it.” More reaction from Richard Bejtlich.

## Martingales from finite Markov processes, part 1

15 February 2010In an earlier series of posts the emerging inhomogeneous Poissonian nature of network traffic was detailed. One implication of this trend is that not only network flows but also individual packets will be increasingly well described by Markov processes of various sorts. At EQ, we use some ideas from the edifice of information theory and the renormalization group to provide a mathematical infrastructure for viewing network traffic as (e.g.) realizations of inhomogeneous finite Markov processes (or countable Markov processes with something akin to a finite universal cover). An essentially equation-free (but idea-heavy) overview of this is given in our whitepaper “Scalable visual traffic analysis”, and more details and examples will be presented over time.

The question for now is, once you’ve got a finite Markov process, what do you do with it? There are some obvious things. For example, you could apply a Chebyshev-type inequality to detect when the traffic parameters change or the underlying assumptions break down (which, if the model is halfway decent, by definition indicates something interesting is going on–even if it’s not malicious). This idea has been around in network security at least since Denning’s 1986-7 intrusion detection article, though, so it’s not likely to bear any more fruit (assuming it ever did). A better idea is to construct and exploit martingales. One way to do this to advantage starting with an inhomogeneous Poisson process (or in principle, at least, more general one-dimensional point processes) was outlined here and here.

Probably the most well-known general technique for constructing martingales from Markov processes is the Dynkin formula. Although we don’t use this formula at present (after having done a lot of tinkering and evaluation), a more general result similar to it will help us introduce the Girsanov theorem for finite Markov processes and thereby one of the tools we’ve developed for detecting changes in network traffic patterns.

The sketch below of a fairly general version of this formula for finite processes is adapted from a preprint of Ford (see Rogers and Williams IV.20 for a more sophisticated treatment).

Consider a time-inhomogeneous Markov process on a finite state space. Let denote the generator, and let denote the corresponding transition kernel, i.e. where the Markov propagator is

and indicates the formal adjoint or *reverse* time-ordering operator. Thus, e.g., an initial distribution is propagated as (NB. Kleinrock‘s queueing theory book omits the time-ordering, which is a no-no.)

Let be bounded and such that the map is Write and Now

and the Markov property gives that

The notation just indicates the history of the process (i.e., its natural filtration) at time The transition kernel satisfies a generalization of the time-homogeneous formula

so the RHS of the previous equation is times

plus a term that vanishes in the limit of vanishing mesh. The fact that the row sums of a generator are identically zero has been used to simplify the result.

Summing over and taking the limit as the mesh of the the partition goes to zero shows that

That is,

is a local martingale, or if is well behaved, a martingale.

This can be generalized (see Rogers and Williams IV.21 and note that the extension to inhomogeneous processes is trivial): if is an inhomogeneous Markov process on a finite state space and is such that is locally bounded and previsible and for all then given by

is a local martingale. Conversely, any local martingale null at 0 can be represented in this form for some satisfying the conditions above (except possibly local boundedness).

To reiterate, this result will be used to help introduce the Girsanov theorem for finite Markov processes in a future post, and later on we’ll also show how Girsanov can be used to arrive at a genuinely simple, scalable likelihood ratio test for identifying changes in network traffic patterns.

## Random bits

12 February 2010## Random bits

10 February 2010Snowstorm round-up edition…

PRC busts a hacker ring…convenient timing for a PR-friendly move. But don’t look too soon…

Mobile phone communication patterns

Graphene superconducting at 90 K

Apparently some people think steganography is nontrivial

## Random bits

4 February 2010Hacking for Fun and Profit in China’s Underworld

Google + NSA Information Assurance Directorate

“Every user in the world is convinced they need security features, not security procedures.”

Advanced Persistent Threat highlighted by DNI; Mandiant report gives details. Mandiant have coined the APT term, and it’s probably because they deal with that sort of thing constantly: they’re very good at what they do. We hired them for internal test and eval work as well as usability input as our software began taking shape, and I came away impressed. It’s not surprising to see them tackling high-profile events.

## Random bits

2 February 2010Congressional Research Service overview of cybersecurity legislation, executive initiatives, and options (PDF)