Secure numerical simulations using fully homomorphic encryption

Data privacy is a significant concern in many environments today. This is particularly true if sensitive information, e.g., engineering, medical, or financial data, is to be processed on potentially insecure systems, as it is often the case in cloud computing. Fully homomorphic encryption (FHE) offers a potential solution to this problem, as it allows for secure computations on encrypted data. In this paper, we investigate the viability of using FHE for privacy-preserving numerical simulations of partial differential equations. We first give an overview of the CKKS scheme, a popular FHE method for computations with real numbers. This is followed by an introduction of our Julia packages OpenFHE.jl and SecureArithmetic.jl, which provide a Julia wrapper for the C++ library OpenFHE and offer a user-friendly interface for secure arithmetic operations. We then present a performance analysis of the CKKS scheme within OpenFHE, focusing on the error and efficiency of different FHE operations. Finally, we demonstrate the application of FHE to secure numerical simulations by implementing two finite difference schemes for the linear advection equation using the SecureArithmetic.jl package. Our results show that FHE can be used to perform cryptographically secure numerical simulations, but that the error and efficiency of FHE operations must be carefully considered when designing applications.