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A Brief Story of Computing on Private Data Ten H Lai Ohio State University

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Agenda Computing on private data Fully homomorphic encryption (FHE) Gentry’s bootstrapping theorem Our result

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FHE: The Holy Grail of Cryptography

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Cloud Computing ServersStoragesNetworksApplications

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天上有多少星星 城裡有多少姑娘 但人間只有一個妳 天上只有一顆月亮

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Cloud Computing 6 Cloud server Internet Encrypt

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Computing on private data 7 Cloud server Internet Encrypt

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Computing on private data Cloud 8 A question proposed by Rivest, Aldeman, Dertouzos in 1978 (one year after RSA was invented).

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C-Homomorphism

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C-homomorphic

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RSA is multiplicatively homomorphic

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Fully Homomorphic Encryption (FHE)

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18 Bootstrapping

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19 m m sk A Decrypt m encrypted under a pink key pk A Evaluate Decrypt m

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m m sk A m m Decrypt Evaluate Decrypt 20 Encrypt under a blue key pk B Evaluate Decrypt

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sk A NAND m 1 NAND m 2 NAND-augmented Decrypt circuit: 21 m1m1 m2m2

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Decrypt sk A c 1 sk A c 2 NAND m 1 NAND m 2 Evaluate 22 fresh m1m1 m2m2

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23 m 1 NAND m 2 23 fresh m1m1 m2m2 sk A Under a pink key PK A Under a blue key PK B

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24 m1m1 m2m2 m 1 NAND m 2 Increased noise

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sk A m 1 m 2 m 1 NAND m 2 Evaluate Decrypt-NAND sk A m 3 m 4 m 3 NAND m 4 Evaluate Decrypt-NAND m 1 NAND m 2 m 3 NAND m 4 Evaluate Decrypt-NAND sk B (m 1 NAND m 2 ) NAND (m 3 NAND m 4 ) 26

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sk A m 1 m 2 m 1 NAND m 2 Evaluate Decrypt-NAND sk A m 3 m 4 m 3 NAND m 4 Evaluate Decrypt-NAND m 1 NAND m 2 m 3 NAND m 4 Evaluate Decrypt-NAND sk B (m 1 NAND m 2 ) NAND (m 3 NAND m 4 ) 27

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28 Decrypt NAND

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30 Encryption key Decryption key Evaluation key

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31 Decrypt

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33 Encryption key Decryption key Evaluation key

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… level d level 1 36

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… Decrypt circuits level d level 1 37

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Decrypt circuits … 38

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41 Encryption key Decryption key Evaluation key

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44 Decrypt NAND

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46 Secret-key independent, Computationally intensive, Done with encryption Secret-key dependent Decryption algorithm

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48 FHE is still in its infantry

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Multi-Key/Multi-Scheme FHE

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Single-key FHE 50

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Is Multi-key FHE Possible? 51

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Is Multi-scheme FHE Possible? 52

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Evaluate circuit C Evaluate(C) Problem

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Eval(C) If under pk 1 C

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Eval(C) Eval( Eval(C) ) Under pk 2 C

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Evaluate(C) ? C

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?

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Trivial encryptions

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Eval(C) Eval( Eval(C) ) Summary of ideas C

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65 C

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