# Homomorphic Lower Digits Removal and Improved FHE Bootstrapping

• Hao Chen
• Kyoohyung Han

EUROCRYPT 2018 |

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))

$\mathrm{log}h+\mathrm{log}\left({\mathrm{log}}_{p}\left(ht\right)\right)$

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

$\mathrm{log}h+2\mathrm{log}t$

to logh+logt

$\mathrm{log}h+\mathrm{log}t$

.

We implemented bootstrapping for FV in the SEAL library. We also introduced another “slim mode”, which restrict the plaintexts to batched vectors in pr

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

. 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)

$GF\left({257}^{128}\right)$

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