BHDR: BSGS-Hoisted Diagonal Regression for Non-Interactive Single-Server Kernel SHAP under CKKS

This technical report introduces BHDR (BSGS-Hoisted Diagonal Regression), a non-interactive single-server construction for computing Kernel SHAP explanations under CKKS fully homomorphic encryption for a deployed logistic-regression path.

The core idea is that the Kernel SHAP sampling matrix Z, kernel weights π, and regression matrix M = (ZᵀWZ)⁻¹ZᵀW are public build-time artefacts. This allows the server to evaluate coalition outputs under encryption and then perform the closing weighted-least-squares regression as a public encrypted matrix-vector product. BHDR implements this regression as a BSGS-hoisted diagonal matvec over a K′-periodic CKKS slot encoding, reducing the encrypted regression rotation count from approximately 2,200 to 51 at d = 50, K = 390, K′ = 512.

The report includes an end-to-end implementation over UCI Adult Income logistic regression, benchmark results, an error-decomposition certificate for deployed releases, and an analytical Matrix Bernstein baseline for an i.i.d. antithetic-pair surrogate sampler. The deployed guarantee is certificate-based: finite-K Kernel SHAP sampling/design error is measured against a high-sample plaintext reference, while CKKS implementation error is bounded through the effective regression-map gain G_eff.

The report also includes a circuit-depth feasibility study for tree ensembles via an Oblivious Coalition Tree Evaluator (OCTE). Tree-ensemble support is presented as a feasibility study; MLP support, malicious-server integrity, and output-privacy hardening are not claimed as part of the main deployed BHDR path.

This release is intended as a public technical report and systems artefact for privacy-preserving explainable machine learning, fully homomorphic encryption, and encrypted feature-attribution research.

Patent notice: The methods described in this technical report are related to pending PCT patent application PCT/IB2026/053405, held by VaultBytes Innovations Ltd. The report is licensed under CC BY 4.0 for text, figures, and tables only. No patent license or implementation right is granted by publication of this report. Implementation or commercial use of patented or patent-pending methods may require a separate license.

Software notice: The reference implementation is distributed separately under AGPL-3.0. The software license applies to the code repository; the publication license for this report applies only to the report text, figures, and tables.