Efficient Bootstrapping in Fully Homomorphic Encryption for Matrix Arithmetic

Recently, Gentry and Lee (GL) proposed a fully homomorphic encryption (FHE) scheme optimized for matrix arithmetic. In this paper, we propose an efficient bootstrapping technique for the GL scheme. Our core idea leverages the linearity of the slot--coefficient transformations, namely CtS and StC: we formulate these operations as ciphertext--plaintext matrix multiplications, which are natively supported by the GL scheme. As a result, the proposed method reduces the number of key-switching operations per step to a small constant. To enable this, we first generalize the GL scheme to matrices of non-power-of-two dimensions by introducing a generalized definition of the trace over commutative rings and proving that it commutes with decryption. Our bootstrapping adopts the CKKS paradigm: ModRaise, CtS, EvalMod and StC. Typically, CtS/StC and EvalMod dominate runtime and depth, respectively; our optimization shifts the bottleneck to EvalMod for both. A proof-of-concept implementation shows that linear transformations account for 20.1% of the total bootstrapping time, compared to 54.9-71.7% in prior CKKS bootstrapping, and that, despite lacking low-level optimizations, our amortized CtS runtime is still about 3 times faster than the well-optimized library (Lattigo).