| Abstract | Topological Data Analysis (TDA) provides a suite of tools that extract shape-based features from high-dimensional data, with applications to modern statistical and machine learning (ML) models. Among these tools, persistent homology (PH) summarizes the topological structure of data in compact representations known as persistence diagrams (PDs). Due to their robustness to noise, interpretability, and compatibility with standard ML architectures, PDs are increasingly used in applications involving sensitive data, such as genomics, cancer research, sensor networks, and finance. Thus, there is a growing need to incorporate TDA methods into secure, end-to-end data analysis pipelines. We present the first adaptation of a fundamental TDA algorithm known as boundary matrix reduction to operate on encrypted data using homomorphic encryption (HE). We provide mathematical guarantees for the correctness of the HE-compatible algorithm under appropriate parameter choices and analyze its computational complexity. We support these theoretical results with two distinct empirical studies: (1) a plaintext simulation that explores the extent to which the theoretically sufficient parameters can be relaxed while still preserving correctness, and (2) a working implementation in the OpenFHE framework that validates correctness on encrypted data. This work lays the foundation for fully encrypted topological computations and opens new directions in privacy-preserving data analysis using TDA. |
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