# A Cryptography-Powered Infrastructure to Ensure the Integrity of Robot Workflows
## 4 Application and Evaluation in a Real Robot System
### Results
#### 4.3.3 Overall System Complexity  
In order to describe the complexity of our solution we take the results presented in this section and interrelate them with the data flow shown in Figure 4. Table 5 contains the complexity of the operations executed in our system in order to sign a workflow queue and verify the resulting signature.  
The first column briefly describes the operation, the second contains the complexity class where \(n\) stands for size of the workflow queue and \(m\) for the number of distinct workflow descriptions. The third column contains the agent/system component where the computational effort is located. Since the operations are executed sequentially and are not nested into each other and \(m<n\), we can conclude that our solution does not exceed linear computational complexity with regards to the workflow queue size.  
While our approach targets larger robot systems that typically have larger computing power, we also want to mention the applicability to embedded robot systems like drones. Since our approach outsources the more expensive cryptographic operations like hashing and signing to the CSP, the robot itself only needs to call the appropriate service. Thus, there is no additional computational on the robot itself except the implementation of the execution controller. If the CSP itself should be realized in an embedded device, we refer the interested reader to a study of cryptography algorithms on restricted  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Operation** & **Complexity** & **Agent** \\ \hline Fetching \(m\) workflow descriptions (if necessary) & \(\mathcal{O}(m)\) & Execution Entity \\ Constructing a WorkflowQueueSigningRequest & \(\mathcal{O}(n)\) & Client \\ Building the Merkle tree & \(\mathcal{O}(n)\) & Compliance Service \\ Signing the root of the tree & \(\mathcal{O}(1)\) & Compliance Service \\ Constructing a WorkflowQueueSignatureVerificationRequest & \(\mathcal{O}(n)\) & Execution Entity \\ Building the Merkle tree & \(\mathcal{O}(n)\) & Compliance Service \\ Verifying the signature against the Merkle tree root & \(\mathcal{O}(1)\) & Compliance Service \\ \hline \hline \end{tabular}
\end{table}
Table 5: Complexity of the overall data flow according to Figure 4. \(n\) is the size of the workflow queue, \(m\) is the number of workflows available on the execution entity.  
Figure 10: Benchmark results.  
devices (in this case smart grids) [45] and the eBACS project ([[https://bench.cr.yp.to/](https://bench.cr.yp.to/)]([https://bench.cr.yp.to/](https://bench.cr.yp.to/))) which provides up-to-date measurements of cryptography on different platforms. We further refer to our groups work on the cryptographic black box which is also implemented on an embedded platform [34]. Finally, we want to point out that the algorithms used by the CSP are contained therein (since client and execution entity just transmit and receive the results). Thus, other cryptographic algorithms can be used in a specific implementation including lightweight algorithms better suited for embedded systems [46].