Bootstrapping is a crucial operation in Gentry\u2019s 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.<\/p>\n\n\n\n
In this work, we applied a family of \u201clowest digit removal\u201d 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\u00a0\\(h=||s||_1\\)<\/p>\n\n\n\n
\\(h=||s||_1\\)<\/p>\n\n\n\n
and the plaintext modulus is\u00a0\\(t=p_r\\)<\/p>\n\n\n\n
\\(t=p^r\\)<\/p>\n\n\n\n
, we achieved bootstrapping depth\u00a0\\(logh+log({log}_p(ht))\\)<\/p>\n\n\n\n
\\(log\u2061h+log\u2061({log}_p\u2061(ht))\\)<\/p>\n\n\n\n
in FV scheme. In case of the BGV scheme, we brought down the depth from\u00a0\\(logh+2logt\\)<\/p>\n\n\n\n
\\(log\u2061h+2log\u2061t\\)<\/p>\n\n\n\n
to\u00a0\\(logh+logt\\)<\/p>\n\n\n\n
\\(log\u2061h+log\u2061t\\)<\/p>\n\n\n\n
.<\/p>\n\n\n\n
We implemented bootstrapping for FV in the SEAL library. We also introduced another \u201cslim mode\u201d, which restrict the plaintexts to batched vectors in\u00a0\\({\\mathbb{Z}}_{p_r}\\)<\/p>\n\n\n\n
\\(Z_{p^r}\\)<\/p>\n\n\n\n
. 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\u00a0s for vectors over\u00a0GF<\/em>(127) with 64 slots and 1381\u00a0s for vectors over\u00a0\\(GF({257}_{128})\\)<\/p>\n\n\n\n \\(GF({257}^{128})\\)<\/p>\n\n\n\n with 128 slots. We also implemented our improved digit extraction procedure for the BGV scheme in HElib.<\/p>\n","protected":false},"excerpt":{"rendered":" Bootstrapping is a crucial operation in Gentry\u2019s 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. 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