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.

翻译：本文评估了不同的全同态加密方案，提出了一种实现方法，并通过数值分析研究了在同态加密设置下使用梯度下降算法求解二次规划的可行性。同态加密电路乘法深度的限制是迭代过程（如梯度下降算法）面临的主要挑战。我们的分析不仅通过原型示例量化了这些限制，从而为未来研究提供了基准，还揭示了额外的权衡问题，例如梯度下降与加速梯度下降方法的选择，为在同态加密技术中广泛应用基于优化的控制中的迭代过程开辟了道路。此外，我们认为，在现有的同态加密方案中，本文采用的CKKS方案是唯一适合实现梯度下降算法的方案。选择适当的步长对于过程的收敛至关重要。本文首次展示了同态加密梯度下降算法的可行性。