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=pr

$t=p^r$

, we achieved bootstrapping depth logh+log(logp(ht))

$\log h+\log ({\log }_p(ht))$

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

$\log h+2\log t$

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 ℤpr

${\mathbb{Z}}_{p^r}$

. 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(257128)

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

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