Snapshot: Fully homomorphic encryption with Julia
In today’s world, data security and privacy are critical, especially when sensitive information is processed on potentially insecure systems, such as in cloud computing. Fully Homomorphic Encryption (FHE) offers a promising solution by enabling secure computations on encrypted data without the need for decryption. Over the past year, Arseniy and Michael of the HPSC Lab have brought to life the idea of performing secure numerical simulations using the Julia programming language. Building upon the efficient C++ implementation of the Cheon-Kim-Kim-Song (CKKS) scheme for real-number arithmetic in the
OpenFHE library, we have created the [OpenFHE.jl](https://github.com/hpsc-lab/OpenFHE.jl) package, which allows the functionalities of OpenFHE to be used from within Julia. Our research group has also introduced the
SecureArithmetic.jl?package, which provides a user-friendly interface for cryptographically secure arithmetic operations. This package includes a backend for OpenFHE-secured computations using OpenFHE.jl, as well as an unencrypted backend for fast verification of computation pipelines. Additionally, SecureArithmetic.jl has expanded the functionality of OpenFHE.jl's vector-based arithmetic operations to support some matrix-based computations. The OpenFHE.jl and SecureArithmetic.jl packages were also the focus of our talk,
Secure numerical computations using fully homomorphic encryption, at JuliaCon 2024 (a YouTube recording of the talk can be found
here). As part of the experimental segment of this talk, we implemented secure versions of well-known finite difference schemes for the linear advection equation, utilizing OpenFHE.jl and SecureArithmetic.jl. The figure below illustrates the results of the 2D linear advection secure simulations at the final time step, with periodic boundary conditions and a sine wave as the initial condition.
Together with Arpit Babbar and Hendrik Ranocha, we have submitted our paper "Automatic differentiation for Lax-Wendroff-type discretizations".
?
?
?
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.