Secure numerical simulations using fully homomorphic encryption

Data privacy is a significant concern when using numerical simulations for sensitive information such as medical, financial, or engineering data—especially in untrusted environments like public cloud infrastructures. Fully homomorphic encryption (FHE) offers a promising solution for achieving data privacy by enabling secure computations directly on encrypted data. Aimed at computational scientists, this work explores the viability of FHE-based, privacy-preserving numerical simulations of partial differential equations. The presented approach utilizes the Cheon-Kim-Kim-Song (CKKS) scheme, a widely used FHE method for approximate arithmetic on real numbers. Two Julia packages are introduced, OpenFHE.jl and SecureArithmetic.jl, which wrap the OpenFHE C++ library to provide a convenient interface for secure arithmetic operations. With these tools, the accuracy and performance of key FHE operations in OpenFHE are evaluated, and implementations of finite difference schemes for solving the linear advection equation with encrypted data are demonstrated. The results show that cryptographically secure numerical simulations are possible, but that careful consideration must be given to the computational overhead and the numerical errors introduced by using FHE. An analysis of the algorithmic restrictions imposed by FHE highlights potential challenges and solutions for extending the approach to other models and methods. While it remains uncertain how broadly the approach can be generalized to more complex algorithms due to CKKS limitations, these findings lay the groundwork for further research on privacy-preserving scientific computing.