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Researchers across various fields seek to understand causal relationships but often find controlled experiments impractical. To address this, statistical tools for causal discovery from naturally observed data have become crucial. Non-linear regression models, such as Gaussian process regression, are commonly used in causal inference but have limitations due to high costs when adapted for secure computation. Support vector regression (SVR) offers an alternative but remains costly in an Multi-party computation context due to conditional branches and support vector updates. In this paper, we propose Aitia, the first two-party secure computation protocol for bivariate causal discovery. The protocol is based on optimized multi-party computation design choices and is secure in the semi-honest setting. At the core of our approach is BSGD-SVR, a new non-linear regression algorithm designed for MPC applications, achieving both high accuracy and low computation and communication costs. Specifically, we reduce the training complexity of the non-linear regression model from approximately from O (𝑁^3) to O (𝑁^2) where 𝑁 is the number of training samples. We implement Aitia using CrypTen and assess its performance across various datasets. Empirical evaluations show a significant speedup of 3.6× to 340× compared to the baseline approach. 

We demonstrate epitaxial lattice-matched Al0.89Sc0.11N/GaN 10 and 20 period distributed Bragg reflectors (DBRs) grown on c-plane bulk n-type GaN substrates by plasma-assisted molecular beam epitaxy. Resulting from a rapid increase in in-plane lattice coefficient as scandium is incorporated into AlScN, we measure a lattice-matched condition to c-plane GaN for a Sc content of just 11%, resulting in a large refractive index mismatch Δn greater than 0.3 corresponding to an index contrast of Δn/nGaN = 0.12 with GaN. The DBRs demonstrated here are designed for a peak reflectivity at a vacuum wavelength of 400 nm, reaching a reflectivity of 0.98 for 20 periods. It is highlighted that AlScN/GaN multilayers require fewer periods for a desired reflectivity than other lattice-matched Bragg reflectors such as those based on AlInN/GaN multilayers.