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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.