  In this paper, we evaluate the different fully homomorphic encryption
schemes, propose an implementation, and numerically analyze the applicability
of gradient descent algorithms to solve quadratic programming in a homomorphic
encryption setup. The limit on the multiplication depth of homomorphic
encryption circuits is a major challenge for iterative procedures such as
gradient descent algorithms. Our analysis not only quantifies these limitations
on prototype examples, thus serving as a benchmark for future investigations,
but also highlights additional trade-offs like the ones pertaining the choice
of gradient descent or accelerated gradient descent methods, opening the road
for the use of homomorphic encryption techniques in iterative procedures widely
used in optimization based control. In addition, we argue that, among the
available homomorphic encryption schemes, the one adopted in this work, namely
CKKS, is the only suitable scheme for implementing gradient descent algorithms.
The choice of the appropriate step size is crucial to the convergence of the
procedure. The paper shows firsthand the feasibility of homomorphically
encrypted gradient descent algorithms.

Cryptography and Security;Systems and Control;Systems and Control