README.md

# Minimal Schedule with Minimal Number of Agents in Attack-Defence Trees

This repository hosts the results for the paper and for its **journal version**.

## Clone this repository:

```
git clone https://depot.lipn.univ-paris13.fr/parties/publications/minimal-scheduling.git && cd minimal-scheduling
```

## Folder Structure

```
.
├── benchmarks # folder with the scripts to generate the benchmarks for adt2amas and adt2maude
├── results # folder with the ADTree models and the minimal assignments
├── script  # folder with the scripts to generate the scalability results
└── tool #  # folder with the binaries of the tool adt2amas
```

## Results

The minimal scheduling results can be found in the `results` folder. The reader
can found the sources of the `adt2amas` tool
[here](https://depot.lipn.univ-paris13.fr/parties/tools/adt2amas/-/releases/v2.1.2),
and the sources of the `adt2maude`
[here](https://depot.lipn.univ-paris13.fr/parties/tools/adt2maude). Binaries for
multiple OS of the `adt2amas` tool can be found in the folder `tool`.

In order to reproduce the results, the command to be executed is the following:

```
# Change adt2amas-linux command by the binary corresponding to your OS.
./tool/adt2amas-linux minimal --print-latex --print-scheduling --model results/<case_study>/model/<case_study>.txt
```

or

```
# adt2maude
./min-schedule.py --table --input results/<case_study>/model/<case_study>.txt
```

Below we show some case studies.

- [Forestalling a software release (forestall)](#forestalling-a-software-release-forestall)
- [Obtain admin privileges (gain-admin)](#obtain-admin-privileges-gain-admin)
- [Interrupted](#interrupted)
- [Compromise IoT device (iot-dev)](#compromise-iot-device-iot-dev)
- [Last](#last)
- [Lukasz](#lukasz)
- [Lukasz-or](#lukasz-or)
- [Example from [5]](#example-from-5)
- [Treasure Hunters](#treasure-hunters)
- [Scaling Example](#scaling-example)

### Forestalling a software release (forestall)

This model is based on a real-world instance [1]. It models an attack
to the intellectual property of a company C, by an unlawful competitor
company U aiming at being “first to the market.” Following [2],
software extraction from C takes place before U builds it into its own
product, and U must deploy to market before C, which takes place after U has
integrated the stolen software into its product.

#### ADTree model

![forestall ADTree](results/forestall/model/forestall.png)

#### Minimal Scheduling

The reader can find the 4 possible assignments [here](results/forestall/assignment).

### Obtain admin privileges (gain-admin)

To gain administrative privileges in a UNIX system, an attacker needs either
physical access to an already logged-in console or remote access via
privilege escalation (attacking SysAdmin). This case study [3] exhibits a
mostly branching structure: all gates but one are disjunctions in the
original tree of [3]. We enrich this scenario with the SAND gates of [2],
and further add reactive defences.

#### ADTree model

![gain-admin ADTree](results/gain-admin/model/gain-admin.png)

#### Minimal Scheduling

The reader can find the 16 possible assignments [here](results/gain-admin/assignment).

### Interrupted

#### ADTree model

![interrupted ADTree](results/interrupted/model/interrupted.png)

#### Minimal Scheduling

![interrupted Assignment](results/interrupted/assignment/interrupted_scheduling_1.png)

### Compromise IoT device (iot-dev)

This model describes an attack to an Internet-of-Things (IoT) device either
via wireless or wired LAN. Once the attacker gains access to the private
network and has acquired the corresponding credentials, it can exploit a
software vulnerability in the IoT device to run a malicious script. Our
ADTree adds defence nodes on top of the attack trees used in [4].

#### ADTree model

![iot-dev ADTree](results/iot-dev/model/iot-dev.png)

#### Minimal Scheduling

The reader can find the possible assignment [here](results/iot-dev/assignment).

### Last

#### ADTree model

![last ADTree](results/last/model/last.png)

#### Minimal Scheduling

![last Assignment](results/last/assignment/last_scheduling_1.png)

### Lukasz

#### ADTree model

![lukasz ADTree](results/lukasz/model/lukasz.png)

#### Minimal Scheduling

![lukasz Assignment](results/lukasz/assignment/lukasz_scheduling_1.png)

### Lukasz-or

#### ADTree model

![lukasz-or ADTree](results/lukasz-or/model/lukasz-or.png)

#### Minimal Scheduling

![lukasz-or Assignment](results/lukasz-or/assignment/lukasz-or_scheduling_1.png)

### Example from [5]

#### ADTree model

![toy-example ADTree](results/toy-example/model/toy-example.png)

#### Minimal Scheduling

![toy-example Assignment](results/toy-example/assignment/toy_example_scheduling_1.png)

### Treasure Hunters

It models thieves that try to steal a treasure in a museum. To achieve their
goal, they first must access the treasure room, which involves bribing a
guard (b), and forcing the secure door (f). Both actions are costly and take
some time. Two coalitions are possible: either a single thief has to carry
out both actions, or a second thief could be hired to parallelise b and f.
After these actions succeed the attacker/s can steal the treasure (ST), which
takes a little time for opening its display stand and putting it in a bag. If
the two-thieves coalition is used, we encode in ST an extra 90 € to hire the
second thief — the computation function of the gate can handle this plurality
— else ST incurs no extra cost. Then the thieves are ready to flee (TF),
choosing an escape route to get away (GA): this can be a spectacular escape
in a helicopter (h), or a mundane one via the emergency exit (e). The
helicopter is expensive but fast while the emergency exit is slower but at no
cost. Furthermore, the time to perform a successful escape could depend on
the number of agents involved in the robbery. Again, this can be encoded via
computation functions in gate GA.

As soon as the treasure room is penetrated (i.e. after b and f but before ST)
an alarm goes off at the police station, so while the thieves flee the police
hurries to intervene (p). The treasure is then successfully stolen iff the
thieves have fled and the police failed to arrive or does so too late. This
last possibility is captured by the condition associated with the treasure
stolen gate (TS), which states that the arrival time of the police must be
greater than the time for the thieves to steal the treasure and go away.

#### ADTree model

![Treasure Hunters ADTree](results/treasure-hunters/model/treasure-hunters.png)

#### Minimal Scheduling

The reader can find the possible assignment [here](results/treasure-hunters/assignment).

### Scaling Example

#### ADTree model

![tricky ADTree](results/tricky/model/tricky.png)

#### Minimal Scheduling

![tricky Assignment](results/tricky/assignment/tricky_scheduling_1.png)

## Scalability Results

We wrote an automatic generator of ADTrees and a notebook that processes the
output of our tool in order to create the figure shown below. The scripts are
detailed in the `script` folder.

The parameters of the generated ADTrees are the _depth_, the _width_
corresponding to the number of deepmost leaves, the number of _children_ for
each AND, and the total number of _nodes_. All nodes have time 1 except the
first leaf that has time _width - 1_. The results show that the number of
agents is not proportional to the width of the tree (red bars - top of
Figure), and the optimal scheduling varies according to the time of nodes
(blue bars - bottom of Figure).

![tricky Assignment](script/scalability.png)

## Authors

- Jaime Arias (LIPN, CNRS UMR 7030, Université Sorbonne Paris Nord)
- Łukasz Maśko (Institute of Computer Science, PAS, Warsaw University of Technology)
- Carlos Olarte (LIPN, CNRS UMR 7030, Université Sorbonne Paris Nord)
- Wojciech Penczek (Institute of Computer Science, PAS, Warsaw University of Technology)
- Laure Petrucci (LIPN, CNRS UMR 7030, Université Sorbonne Paris Nord)
- Teofil Sidoruk (Institute of Computer Science, PAS, Warsaw University of Technology)

## Abstract

Expressing attack-defence trees in a multi-agent setting allows for studying a
new aspect of security scenarios, namely how the number of agents and their task
assignment impact the performance, e.g. attack time, of strategies executed by
opposing coalitions. Optimal scheduling of agents' actions, a non-trivial
problem, is thus vital. We discuss associated caveats and propose an algorithm
that synthesises such an assignment, targeting minimal attack time and using the
minimal number of agents for a given attack-defence tree. We also investigate an
alternative approach for the same problem using Rewriting Logic, starting with a
simple and elegant declarative model, whose correctness (in terms of schedule's
optimality) is self-evident. We then refine this specification, inspired by the
design of our specialised algorithm, to obtain an efficient system that can be
used as a playground to explore various aspects of attack-defence trees. We
compare the two approaches on different benchmarks.

## References

[1] A. Buldas, P. Laud, J. Priisalu, M. Saarepera, and J. Willemson. Rational
Choice of Security Measures Via Multi-parameter Attack Trees. In Critical
Information Infrastructures Security, pages 235–248. Springer, 2006.

[2] R. Kumar, E. Ruijters, and M. Stoelinga. Quantitative attack tree analysis
via priced timed automata. In FORMATS 2015, volume 9268 of LNCS, pages 156– 171. Springer, 2015.

[3] J. D. Weiss. A system security engineering process. In Proceedings of the
14th National Computer Security Conference, pages 572–581, 1991.

[4] R. Kumar, S. Schivo, E. Ruijters, B. M. Yildiz, D. Huistra, J. Brandt, A.
Rensink, and M. Stoelinga. Effective analysis of attack trees: A model-driven
approach. In Fundamental Approaches to Software Engineering, pages 56–73.
Springer, 2018.

[5] J. Arias, C. E. Budde, W. Penczek, L. Petrucci, T. Sidoruk, and M.
Stoelinga. Hackers vs. Security: Attack-Defence Trees as Asynchronous
Multi-agent Systems. In Formal Methods and Software Engineering, vol. 12531,
pages 3-19. Springer, 2020.