Approximate Decryption in Homomorphic Division and Privacy Impact

Abstract

The result of a computation over secret inputs inherently reveals some information about those inputs; such semantic leakage is unavoidable. The challenge is to ensure that the computation method does not introduce additional, avoidable disclosure beyond what is implied by the output itself. This issue is particularly critical in privacy-preserving machine learning and cloud-based data processing, where homomorphic encryption enables computation over encrypted data but often relies on practical approximations.

Division-enabled homomorphic encryption schemes based on rational encodings preserve arithmetic correctness, but their decrypted outputs may retain sufficient algebraic structure to enable inference of the original operands, creating representation-induced leakage.

We study approximate decryption as a privacy-aware interpretation mechanism for homomorphic division, by combining symmetric additive masking with continued fraction expansion to recover meaningful approximations while avoiding exact reconstruction. We empirically compare Shared-k and Distinct ka,kb with respect to numerical growth and reconstruction accuracy under approximate decryption, showing that the latter achieves smoother growth and lower reconstruction failure. This work identifies a previously underexplored privacy risk and demonstrates that approximation-based decryption provides a practical mitigation in settings where bounded numerical error is acceptable.