FHE for SIMD Arithmetic Logic Units with Amortized O(1) Bootstrapping per Ciphertext

The evaluation of both arithmetic and logic operations on machine words (e.g., 64-bit registers) in homomorphic settings is an attractive problem due to its close alignment with real-world programming models. Existing FHE schemes require iterative bootstrapping operations with the iteration number scaling with the machine word bit-width $n$. Prior approaches incur either quadratic cost for multiplication (DM/CGGI), linear cost for logic operations (REFHE [Eurocrypt'26] and Kim [TCHES'26]), or logarithmic cost for multiplication (CPL [Eurocrypt'26]).

We mitigate this inherent barrier by amortizing $O(n)$ iterations across $O(n)$ ciphertexts for scenarios with sufficient machine words. We introduce a triangle encoding such that in its conversion to boolean mode, each iteration on a combined ciphertext processes multiple ciphertexts, resulting in an amortized constant cost per ciphertext. Additionally, the triangle encoding supports leveled arithmetic, and its refreshing requires only two CKKS bootstrapping operations. We reuse discrete CKKS as our boolean mode, which supports lightweight conversion to the triangle encoding. We also introduce a method to induce SIMD structure for the triangle encoding from the RLWE ring.

Even when we bootstrap after each multiplication, we show a 1.7$\times$ to 2$\times$ higher throughput compared with the state-of-the-art CPL scheme. We further estimate 15$\times$ and 491$\times$ higher throughput for multiplication and bitwise operations, respectively, compared with the REFHE scheme. We make our OpenFHE-based prototype open source for ease of reference.