Converting Boolean-Logic Decision Trees to Finite State Machines
for simpler, high-performance detection of cybersecurity events
When analyzing cybersecurity events, the detection algorithm evaluates attributes against boolean expressions to determine whether the event belongs to a class. This article describes converting boolean expressions to finite state machines to permit simpler, high-performance evaluation.
The open-source project Cyberprobe features this implementation. Conversion of rules to finite state machine (FSM) and application of the rules in FSM form is implemented in Python. Cyberprobe supports the use of millions of rules, which can be applied at greater than 200k events/second on a single processor core.
Applying boolean logic criteria to events solves many scanning and detection problems. For instance, an event occurs that is generated from an interaction with a service under protection. The event has the following attributes:
- Source address:
- Destination address:
One or more boolean expressions for the class of thing I am trying to detect:
If TCP port number is 80 or 8080 AND IP address is 10.0.0.1 AND URL is http://www.example.com/malware.dat OR http://example.com/malware.dat …
The aim is to analyze a high-rate stream of such events against a large set of boolean expressions to classify the events.
The boolean expressions get unreadable quickly with English, which has no built-in operator precedence.
Boolean operators are represented as functions, and
type:value represents attribute type/value match terms.
A boolean expression consists of a combination of,
not(…) functions, along with
type:value match terms. I am using
type:value pairs for match terms as that is useful in the domain I’m working in, but we could just as easily use strings.
When evaluating the attributes of an event, attributes are
type:value pairs. e.g.
A basic evaluation algorithm
A simple approach for evaluation of a boolean expression using
type:value pair input is to represent the boolean expression as a tree, and then use
type:value pairs to trigger evaluation. Observations are stored in the tree.
The rules for evaluating a boolean tree against an event are:
- For each
type:valueattribute, see if there is a corresponding
type:valueterm in the boolean tree. If it exists, set the term node as true, and evaluate the parent node.
- When evaluating a parent or node, when any child is true, the or node is true, and its parent node is evaluated.
- When evaluating a parent and node, when ALL children are true, the and node is true, and its parent node is evaluated.
- When evaluating a parent not node, when the child node is true, the not node is false. Once evaluation of all attributes is complete, if a not node has not been deemed false because its child is false, then it is evaluated true, and it’s parent node is evaluated.
That’s a straightforward algorithm; the point of this article is to provide an optimization.
There is a compromise here, the algorithm to convert the boolean tree to an FSM is compute intensive: it has complexity which is non-linear with the number of nodes: it is linear with the product of combination nodes (described below) and
type:value terms. In real-world scenarios, boolean expressions will be converted to FSM when the rule is parsed, thereafter the FSM can be used numerous times.
Converting to an FSM
Step 1: Identify the ‘basic states’
In order to find the FSM, we look for all of the nodes in the boolean tree where state needs to be observed as evaluation proceeds. If you look at the example above, you can see that or nodes and and nodes are different. A child of an or node when evaluated as true immediately results in its parent being true, so no state needs to be kept regarding the children of or nodes. Whereas, when a child of an and node is true this is something which may need to be stored for later evaluation to determine the point at which the and node can be evaluated true.
The evaluation of not nodes is also complicated: a not node can be evaluated as true by virtue of its child maintaining a false evaluation for the duration of analysis.
The rules we state here are that some nodes in the boolean tree can be described as basic states:
- The root of a tree is inherently a
hitstate, which means the boolean expression is true. This is a basic state.
- A not node is never a basic state.
- A child of an and node is a basic state unless it is a not node.
- A child of a not node is a basic state unless it is a not node itself.
In the above example, the basic states are the two or nodes, and the
ip:10.0.0.1 node. All qualify under rule 3.
The implementation gives each state a state name which consists of the letter s plus a unique number, assigned in a depth-first walk. The example boolean tree with states is shown below; the three children of the and node are given states, with the parent and node representing the hit state.
Step 2: Identify the ‘combination states’
The basic states are nodes where partial state needs to be recorded. One node in an FSM represents all state at the same time i.e. all the valid basic state combinations. Hence the combination states set consists all combinations of basic states. This includes the empty set, and a union of all states.
Combination states need to have a state name: in my implementation, I combine states to a name by ordering, separating state numbers with a hyphen preceded by
s. For example, a combination of states
s13 is called
The empty set has a special name which we call
init. It represents the initial state of the FSM where no information is known.
There is a special state
hit which is used to describe any combination of basic states which include the root node evaluating to true. The combination of other states is ignored.
In the above example, the combination state set consists of:
init: The empty set
s3: The first or node:
s7: The second or node
s3-4: The first or node and
ip:10.0.0.1node and the second or node
s3-7: The first and second or nodes
hit: the root node
Step 3: Find all match terms
This is the set of all
type:value match nodes in the boolean expression tree.
Step 4: Find all transitions
This step is essentially about working out what all
type:value match nodes do to all combination states. There is a special match term,
end: which is used to evaluate what happens to not nodes when the list of terms is completed.
The algorithm is:
For every combination state:
Work out the state name of that 'input' combination state
For every match term:
Given the input state
What state results from evaluating that term as true?
Work out the state name of that 'output' combination state
Record a transition (input, match term, output)
Given the input state
What state results from evaluating end: as true?
Work out the state name of that 'output' combination state
Record a transition (input, end:, output)
For this analysis, when the whole boolean expression evaluates as true i.e. the root node of the boolean expression is true, we give that a special name
The result is a complete set of triples: (input, term, output). If the input and output states are the same, we can ignore the transition so that the FSM only contains edges which change state.
At this point, the FSM has some inefficiencies: there may be areas of the FSM which it is not possible to navigate to from
init. This is addressed in the next step.
Step 5: Remove invalid transitions
Not all combination states can be reached from
init, and so some of the transitions discovered can be discarded as irrelevant.
We start by constructing a set of states which can navigate to
- Create a set containing only the combination state
- Iterate over the FSM adding all transitions for which there is a navigation to any state in the set.
- Repeat 2. until the full set of states is discovered.
At this point we know all states which can lead to
hit. However, there will be transitions which lead to states which are not in this set, and thus cannot ever travel to
hit. So, the first simplification of invalid transitions is to reduce all transitions to states which are NOT in this set to the single state named
There is a second simplification of the FSM: some of the states are not navigable from
init, and can be removed:
- Construct a set containing only
- Iterate over the FSM finding all transitions for which there is a navigation from any state in the set.
- Repeat 2. until the set of states is discovered.
At this point we know areas of the FSM which are not reachable, and they can be removed.
The FSM of the above binary tree is depicted below. The
init state represents the initial FSM state. The
hit state represents successful evaluation of the boolean expression as true. We have mentioned the
fail state, which only occurs when not expressions are used, which do not appear as a result of the boolean expression described as above. See below for an example.
Using the FSM
Evaluation of a boolean expressions using the FSM is simple:
- The FSM starts in the
- As attributes are discovered, the
type:valueis compared to the transitions from the current state. If a transition exists, the FSM moves to a new state.
- When the
hitstate is achieved, that is the equivalent of the boolean expression evaluating to true.
- When the
failstate is achieved, no further attribute discovery is needed, and the evaluation can be fast-failed.
The fact that for each term a single FSM lookup is needed means that this approach has performance advantages.
Example 2: Using not
For this example, the not node is used:
It is interesting to view this graph before removal of invalid transitions and discovery of fail states. Some example artifacts:
- There is no transition to the combination state consisting only of state
s9. This is because it is not possible to arrive at this state without evaluating both
s8as true. There are transitions that lead from
s9, to hit, but there is no path which leads to the
s9state, so they can never be taken.
s5-8is similarly not valid, if
s8are evaluated as true,
s9is also true. In both cases, the valid state for this condition would be
s5-8-9. This results in an unreachable part of the FSM with two nodes which is not connected with the rest of the FSM.
- Any state with
s3true cannot lead to
hitbecause the root and node is necessarily false. All nodes which include
s3can be replaced by the
After removal of invalid transitions and mapping to fail states, the FSM is easier to understand:
This example illustrates the
fail state: once transitions lead to this state, it is not possible for further information to permit transition to the
hit state. Discovering
tcp:8082 to be present in any state causes a transition to the
fail state. The fail state could be useful depending on your analysis strategy: it may be a point to fast-fail and shortcut further evaluation. This example also illustrates the special
end: term which leads to
Example 3: More state
This example requires much more state to be kept as a result of all of the and conditions.
The resultant FSM has many states as a result:
Evaluating many rules concurrently
The FSM model lends itself to highly performant scanning using a large set of rules, each converted to an FSM using the approach described above. While it is theoretically possible to produce an uber-FSM from the individual FSMs, the size of the FSM rapidly becomes unwieldly. As there needs to be a single state in the uber-FSM for each combination of states in the set of contributary FSMs.
However, tracking a large number of FSM states concurrently could be compute-intensive.
A simple evaluation approach is to identify a set of initiators, which are the set of terms which lead away from the
init state in each FSM. If any of the initiators are detected while analyzing attributes, the corresponding FSM is activated and put on the set of FSMs which are being tracked for state changes in subsequent match term evaluation. This approach reduces the number of FSMs which need to be tracked. I find in practice, this results in a small set of FSMs used for evaluation.
Using this approach, it is not appropriate to fast-fail an FSM and remove it from the set of tracked FSMs; FSMs must be tracked to the fail state, to prevent the FSM from subsequently being re-activated.
Implementation: cyberprobe indicators
The cyberprobe project includes a means to write rules in JSON format. There are a number of utilities which parse the rule format and output FSM information e.g.
indicators-show-fsm takes a rule/indicator file and dumps out the FSMs of every rule in the file. This is output in human-readable form showing the state transitions:
[indicators]$ indicators-show-fsm case1.json
3ce77704-abe4–4527–84e6-ed6a745aebcf: URL of a page serving malware
init — tcp:8080 -> s6
init — tcp:80 -> s6
init — url:http://example.org/malware.dat -> s3
init — url:http://www.example.org/malware.dat -> s3
s3 — tcp:8080 -> hit
s3 — tcp:80 -> hit
s6 — url:http://example.org/malware.dat -> hit
s6 — url:http://www.example.org/malware.dat -> hit
indicators-graph-fsm is a utility which takes a rule/indicator file and a rule ID, and outputs a Graphviz format graph describing the FSM, which is how I generated the diagrams in this article:
[indicators]$ indicators-graph-fsm case1.json \
3ce77704-abe4–4527–84e6-ed6a745aebcf > graph.dot
[indicators]$ dot -Tpng graph.dot > graph.png
indicators-dump-fsm is a utility which takes an indicator file and outputs the FSM in a JSON form.
You can embed the FSM transformation in your code using the
#!/usr/bin/env python3import sys
import cyberprobe.fsm_extract as fsme
from cyberprobe.logictree import And, Or, Not, Matchexpression = And([
Match("tcp", "80"), Match("tcp", "8080")
])fsm = fsme.extract(expression)# Dump out FSM
for v in fsm:
for w in v:
print(" %s -- %s:%s -> %s" % (v, w, w, v))
A quick word on performance
To benchmark my algorithm, I have compared the performance of the FSM approach with the basic boolean-tree algorithm discussed at “A basic algorithm” above, coding both in Python. The plot below shows the number of rules in use as the x-axis, and the event handling rate as the y-axis. This is a way to show how the number of rules in use affects event throughput. The orange line represents the performance of the FSM. You can see that the number of rules in use has very little affect on algorithm throughput. Note logarithmic scale on the x-axis.
This was run on VirtualBox on my old MacBook, expect better performance from a cloud VM, for instance.
That’s a very quick look at performance, maybe I’ll do a follow-up article on performance later.
In this article I have discussed:
- How boolean expressions can be represented as trees
- Mapping boolean expressions to finite state machines
- How the use of an FSM simplifies detection logic and enables performance advantages
- How to use multiple FSMs for evaluation of concurrent boolean expressions
- That the FSM approach is implemented in the open source cyberprobe project