Dyer’s vs BFV

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Disclosure of Funding Source: This research was conducted at the Georgia Institute of Technology under the supervision of Dr. Jun Ueda, completing in November 2022, resulting in the following publication: Secure Teleoperation Control Using Somewhat Homomorphic Encryption.

This study was supported in part by National Science Foundation Grant Nos. 2112793. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

This project was the first of my graduate studies, and developed foundational ideas used in the remainder of my graduate research projects. As such, it is recommended to read this page before the other project pages.

Encrypted Controller

An encrypted controller is a digital controller running inside the encrypted-space of a homomorphic cipher–a type of cipher that allows computation on encrypted data. Given the great computational burden imposed by encrypted calculation and the real-time requirements of digital controllers, encrypted controllers are extremely difficult to construct.

This research dives head first into this challenge and attempts to construct a digital controller using Somewhat Homomorphic Encryption (SHE)–a class of ciphers that support both addition and multiplication but has a finite number of allowed operations before the data corrupts.

Using SHE we can completely conceal all sensor signals, model parameters, feedback gains, and perform computations in the ciphertext space to generate motion commands without a security hole. This allows us to keep every aspect of our control system secret from a cloud provider hosting our controller.

As of time of writing most of the encrypted controller designs proposed in the academic literature have used so called Partially Homomorphic Encryption (PHE), which only allows either multiplication or addition but not both.

PHE Controller

PHE ciphers such as RSA (multiplicative), ElGamal (multiplicative), Paillier (additive) have been around for awhile and are fast enough under the single encrypted operation they offer. Because of this they have been the primary ciphers of study in encrypted controls research.

Since a PHE cipher can only perform a single operation the other operation must be done at the plant. Typically the multiplications are performed in the cloud and the additions are performed plant-side. This however leaves partial information exposed when the plant decrypt the received products from the cloud. The result is that information about the controller’s operations are leaked plant-side when using PHE.

This information leak from using PHE is depicted below:

This research seeks to address the security holes offered by PHE based encrypted controllers by constructing a SHE based encrypted controller. Note that SHE in general is also known to be computationally expensive and its application to real-time control is considered to be infeasible. However, this and the other research projects which compose my MS thesis, will show that with proper parameter selection and accompanying support circuits, SHE based encrypted control is possible.

SHE Controller

The use of SHE in encrypted control systems has been considered impossible in past years due to their operation limit and slow computation time, however this research shows the feasibility of a SHE encrypted controller.

Since SHE supports both addition and multiplication we do not need to worry about information leak as we did in the PHE based system. This is because all operations (i.e. both addition and multiplication) are being done in cipher space, so no intermediate computation is revealed.

We can see the increased security configuration offered by SHE in the diagram below:

While SHE solves our information leak problem it imposes its own technical challenges which must be overcome. Since the number of operations are limited we must take great care to make sure our control equation can be computed within the allowed operation budget, something addressed more in another project.

Teleoperation

The viability of a real-time encrypted controller was tested with simulations of a PD teleoperation control system. Such a system has two physical systems, known as plants, virtually coupled together via a distributed control scheme taking shared information from each plant. A typical teleoperation configuration involves an operator controlling a remote manipulator.

This research then investigates the use of different homomorphic ciphers to encrypt the shared information sent by each plant to the cloud controller. This allows the cloud to securely compute control commands for both plants, without reveling any plant information to the cloud. Such a system is depicted in the figure below:

The challenge in realizing such a system lies in overcoming the significant computation burden homomorphic encryption imposes so to reach realtime performance. Overcoming this difficulties is one of the objectives of this research.

Numerical simulations were conducting between an integer based homomorphic cipher and a Ring Learning With Error (R-LWE) based cipher. The timing results of these simulations are shown in the following bar graphs:

Conclusion

Overall this research proposed a novel somewhat homomorphic cipher based controller as a potential solution to realize real-time encrypted robotic control. The study went on to compare different classes of ciphers to determine which was best suited to achieve target performance.