New submission: Secure numerical simulations using fully homomorphic encryption
Together with
Yuriy Polyakov from Duality Technologies Inc. we have submitted our paper "Secure numerical simulations using fully homomorphic encryption". Thank you very much, Yuriy, for the productive and enjoyable collaboration on this project! ? 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.
Abstract
Together with Arpit Babbar and Hendrik Ranocha, we have submitted our paper "Automatic differentiation for Lax-Wendroff-type discretizations".
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Abstract
Lax-Wendroff methods combined with discontinuous Galerkin/flux reconstruction spatial discretization provide a high-order, single-stage, quadrature-free method for solving hyperbolic conservation laws. In this work, we introduce automatic differentiation (AD) in the element-local time average flux computation step (the predictor step) of Lax-Wendroff methods. The application of AD is similar for methods of any order and does not need positivity corrections during the predictor step. This contrasts with the approximate Lax-Wendroff procedure, which requires different finite difference formulas for different orders of the method and positivity corrections in the predictor step for fluxes that can only be computed on admissible states. The method is Jacobian-free and problem-independent, allowing direct application to any physical flux function. Numerical experiments demonstrate the order and positivity preservation of the method. Additionally, performance comparisons indicate that the wall-clock time of automatic differentiation is always on par with the approximate Lax-Wendroff method.