PaCo: Bootstrapping for CKKS via Partial CoeffToSlot

We introduce PaCo, a novel and efficient bootstrapping procedure for the CKKS homomorphic encryption scheme, where PaCo stands for “(Bootstrapping via) Partial CoeffToSlot”. At a high level, PaCo reformulates the CKKS decryption equation in terms of blind rotations and modular additions. This reformulated decryption circuit is then evaluated homomorphically within the CKKS framework. Our approach makes use of the circle group in the complex plane to simulate modular additions via complex multiplication, and utilizes alternative polynomial ring structures to support blind rotations. These ring structures are enabled by a variant of the CoeffToSlot operation, which we call a partial CoeffToSlot. This yields a new bootstrapping approach within CKKS, achieving a computational complexity which is logarithmic in the number of complex slots. We further introduce a parallelized variant that enables bootstrapping over all CKKS slots with enhanced throughput, highlighting PaCo’s suitability for practical and large-scale homomorphic applications. In addition to the bootstrapping technique itself, we develop several supporting tools — particularly in the context of bit-reversing and alternative ring structures for CKKS — which can be of independent interest to the community. Finally, a proof-of-concept implementation confirms that PaCo achieves performance competitive with state-of-the-art methods for CKKS bootstrapping.