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

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Presentation transcript:

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

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

FHE: The Holy Grail of Cryptography

Cloud Computing ServersStoragesNetworksApplications

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

Cloud Computing 6 Cloud server Internet Encrypt

Computing on private data 7 Cloud server Internet Encrypt

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

C-Homomorphism

C-homomorphic

RSA is multiplicatively homomorphic

Fully Homomorphic Encryption (FHE)

15

16

17

18 Bootstrapping

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

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

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

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

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

24 m1m1 m2m2 m 1 NAND m 2 Increased noise

25

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

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

28 Decrypt NAND

29

30 Encryption key Decryption key Evaluation key

31 Decrypt

32

33 Encryption key Decryption key Evaluation key

34

35

… level d level 1 36

… Decrypt circuits level d level 1 37

Decrypt circuits … 38

39

40

41 Encryption key Decryption key Evaluation key

42

43

44 Decrypt NAND

45

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

47

48 FHE is still in its infantry

Multi-Key/Multi-Scheme FHE

Single-key FHE 50

Is Multi-key FHE Possible? 51

Is Multi-scheme FHE Possible? 52

53

54

55

56

Evaluate circuit C Evaluate(C) Problem

Eval(C) If under pk 1 C

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

Evaluate(C) ? C

?

62

Trivial encryptions

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

65 C

69