CHLOE: Loop Transformation over Fully Homomorphic Encryption via Multi-Level Vectorization and Control-Path Reduction

This work proposes a multi-level compiler framework to transform programs with loop structures to efficient algorithms over fully homomorphic encryption (FHE). We observe that, when loops operate over ciphertexts, it becomes extremely challenging to effectively interpret the control structures within the loop and construct operator cost models for the main body of the loop. Consequently, most existing compiler frameworks have inadequate support for programs involving non-trivial loops, undermining the expressiveness of programming over FHE. To achieve both efficient and general program execution over FHE, we propose CHLOE, a new compiler framework with multi-level control-flow analysis for the effective optimization of compound repetition control structures. We observe that loops over FHE can be classified into two categories depending on whether the loop condition is encrypted, namely, the transparent loops and the oblivious loops. For transparent loops, we can directly inspect the control structures and build operator cost models to apply FHE-specific loop segmentation and vectorization in a fine-grained manner. Meanwhile, for oblivious loops, we derive closed-form expressions and static analysis techniques to reduce the number of potential loop paths and conditional branches. In the experiment, we show that CHLOE can compile programs with complex loop structures into efficient executable codes over FHE, where the performance improvement ranges from 1.5× to 54× (up to 10⁵× for programs containing oblivious loops) when compared to programs produced by the-state-of-the-art FHE compilers.