Homomorphic Lower Digits Removal and Improved FHE Bootstrapping

Bootstrapping is a crucial operation in Gentry’s breakthrough work on fully homomorphic encryption (FHE), where a homomorphic encryption scheme evaluates its own decryption algorithm. There has been a couple of implementations of bootstrapping, among which HElib arguably marks the state-of-the-art in terms of throughput, ciphertext/message size ratio and support for large plaintext moduli.

In this work, we applied a family of “lowest digit removal” polynomials to design an improved homomorphic digit extraction algorithm which is a crucial part in bootstrapping for both FV and BGV schemes. When the secret key has 1-norm \(h=||s||_1\)

\(h=||s||_1\)

and the plaintext modulus is \(t=p_r\)

\(t=p^r\)

, we achieved bootstrapping depth \(logh+log({log}_p(ht))\)

\(log⁡h+log⁡({log}_p⁡(ht))\)

in FV scheme. In case of the BGV scheme, we brought down the depth from \(logh+2logt\)

\(log⁡h+2log⁡t\)

to \(logh+logt\)

\(log⁡h+log⁡t\)

.

We implemented bootstrapping for FV in the SEAL library. We also introduced another “slim mode”, which restrict the plaintexts to batched vectors in \({\mathbb{Z}}_{p_r}\)

\(Z_{p^r}\)

. The slim mode has similar throughput as the full mode, while each individual run is much faster and uses much smaller memory. For example, bootstrapping takes 6.75 s for vectors over GF(127) with 64 slots and 1381 s for vectors over \(GF({257}_{128})\)

\(GF({257}^{128})\)

with 128 slots. We also implemented our improved digit extraction procedure for the BGV scheme in HElib.

Opens in a new tab