Paper 2026/975

Functional Bootstrapping for a Single LWE Ciphertext with \(\tilde{O}(1)\) Polynomial Multiplications

Abstract

Bootstrapping is the key technique that turns leveled homomorphic encryptionc into fully homomorphic encryption, but it remains a major efficiency bottleneck. Recent work by Z. Liu and Y. Wang (ASIACRYPT 2023) showed how to bootstrap \(N\) LWE ciphertexts with total cost of \(\widetilde{O}(N)\) polynomial multiplications based on the BFV scheme. However, their results achieve \(\widetilde{O}(1)\) complexity only through amortization over large batches, and do not give a genuine non-amortized \(\widetilde{O}(1)\) bound for a single ciphertext. In this paper, we present a BFV-based functional bootstrapping algorithm for arbitrary functions over large plaintext spaces with total cost of \(\widetilde{O}(1)\) polynomial multiplications for one LWE ciphertext. The same construction also supports small and moderate batches, and processes a batch of \(m\) ciphertexts with total cost \(\widetilde{O}(m)\) in the supported parameter range. The main technical ingredient is a sparse-packing polynomial-evaluation method for BFV ciphertexts, which exploits the duplicated-slot structure to evaluate an arbitrary polynomial on \(m\) encrypted inputs with total cost of \(\widetilde{O}(m)\). We implement the scheme in Lattigo using the BFV scheme. At 128 bit security and on a single thread, bootstrapping an arbitrary function takes 3.15 seconds for one ciphertext encrypting a 9-bit plaintext and 3.77 seconds for 128 such ciphertexts in one batched invocation. For 16-bit plaintexts, it takes 10.63 seconds for one ciphertext and 18.07 seconds for 16 ciphertexts. These results show that non-amortized single-ciphertext functional bootstrapping, as well as small and moderate batch bootstrapping, can be practical for arbitrary functions over relatively large plaintext spaces.