{"id":498032,"date":"2018-07-27T14:20:44","date_gmt":"2018-07-27T21:20:44","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=498032"},"modified":"2018-10-16T20:13:29","modified_gmt":"2018-10-17T03:13:29","slug":"homomorphic-lower-digits-removal-and-improved-fhe-bootstrapping-3","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/homomorphic-lower-digits-removal-and-improved-fhe-bootstrapping-3\/","title":{"rendered":"Homomorphic Lower Digits Removal and Improved FHE Bootstrapping"},"content":{"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.<\/p>\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\u00a0h<\/span>=<\/span>|<\/span><\/span><\/span>|<\/span><\/span><\/span>s<\/span>|<\/span><\/span><\/span>|<\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

h<\/mi>=<\/mo>|<\/mo><\/mrow>|<\/mo><\/mrow>s<\/mi>|<\/mo><\/mrow>|<\/mo><\/mrow>1<\/mn><\/msub><\/math><\/p>\n

and the plaintext modulus is\u00a0t<\/span>=<\/span>p<\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

t<\/mi>=<\/mo>p<\/mi>r<\/mi><\/msup><\/math><\/p>\n

, we achieved bootstrapping depth\u00a0log<\/span><\/span>h<\/span>+<\/span>log<\/span><\/span>(<\/span>log<\/span>p<\/span><\/span><\/span>(<\/span>h<\/span>t<\/span>)<\/span>)<\/span><\/span><\/span><\/span><\/span><\/p>\n

log<\/mi>\u2061<\/mo>h<\/mi>+<\/mo>log<\/mi>\u2061<\/mo>(<\/mo>log<\/mi>p<\/mi><\/msub>\u2061<\/mo>(<\/mo>h<\/mi>t<\/mi>)<\/mo>)<\/mo><\/math><\/p>\n

in FV scheme. In case of the BGV scheme, we brought down the depth from\u00a0log<\/span><\/span>h<\/span>+<\/span>2<\/span>log<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span><\/p>\n

log<\/mi>\u2061<\/mo>h<\/mi>+<\/mo>2<\/mn>log<\/mi>\u2061<\/mo>t<\/mi><\/math><\/p>\n

to\u00a0log<\/span><\/span>h<\/span>+<\/span>log<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span><\/p>\n

log<\/mi>\u2061<\/mo>h<\/mi>+<\/mo>log<\/mi>\u2061<\/mo>t<\/mi><\/math><\/p>\n

.<\/p>\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\u2124<\/span><\/span><\/span>p<\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

Z<\/mi><\/mrow>p<\/mi>r<\/mi><\/msup><\/mrow><\/msub><\/math><\/p>\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\u00a0G<\/span>F<\/span>(<\/span>257<\/span>128<\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/p>\n

G<\/mi>F<\/mi>(<\/mo>257<\/mn>128<\/mn><\/mrow><\/msup>)<\/mo><\/math><\/p>\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. In this […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"EUROCRYPT 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