Meta-PBS: Compact High-Precision Programmable Bootstrapping

Currently, most FHE schemes realize bootstrapping through the linear-decrypt-then-round paradigm. For the programmable bootstrapping (PBS) of TFHE, this means the lookup table (LUT) needs a redundancy of $O(\sqrt{N})$ to be able to remove the modulus switching noise, which limits the plaintext modulus of PBS to $O(\sqrt{N})$. We remove this requirement for redundancy by proposing the Meta-PBS framework, which allows us to start with under-redundant or non-redundant LUTs. Meta-PBS iteratively blind-rotates the LUT, during which the LUT redundancy gradually increases. The bootstrapping outputs the correct result when the redundancy eventually exceeds the noise bound. Asymptotically, Meta-PBS requires $O(1)$ blind rotations in dimension $N$ to evaluate a negacyclic function modulo $2N$, whereas PBS needs $O(\sqrt{N})$ blind rotations. Meta-PBS also enjoys an additive noise growth, allowing for more homomorphic arithmetic on bootstrapped ciphertext. We modified Meta-PBS to support the simultaneous evaluation of multiple LUTs on the same ciphertext and/or arbitrary LUTs. According to our implementation, when evaluating a 12-bit negacyclic function, Meta-PBS outperforms EBS (PKC'23) by 79 times. When evaluating an arbitrary function on an 8-bit LWE ciphertext, Meta-PBS reduces the running time of the Refined LHE (CCS'25) by half while allowing for a 27 times larger post-bootstrap linear combination.