Fiat-Shamir and correlation intractability from strong kdm secure encryption
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1 Fiat-Shamir and correlation intractability from strong kdm secure encryption Ran Canetti, Yilei Chen, Leonid Reyzin, Ron Rothblum Eurocrypt 2018, Tel Aviv 1
2 2
3 How are you? 3
4 How are you? Great! How are you? 4
5 How are you? Great! How are you? Great! 5
6 How are you? Great! How are you? Great! Have a great day! A typical 4-message greeting protocol in America 6
7 Israeli 7
8 What s up? Israeli 8
9 What s up? Not bad. Israeli 9
10 What s up? Not bad. Israeli Round reducing, fast and clean! 10
11 Fiat, Shamir 86: 3 round interactive protocol => 1 round argument 11
12 P a b (public coins) c V Fiat, Shamir 86: 3 round interactive protocol => 1 round argument 12
13 P a b (public coins) c V => P a, b=h(a), c V Fiat, Shamir 86: 3 round interactive protocol => 1 round argument 13
14 P a b (public coins) c V => P a, b=h(a), c V Fiat, Shamir 86: 3 round interactive protocol => 1 round argument Pointcheval, Stern 96: secure in the random oracle model. 14
15 P a b (public coins) c V => P a, b=h(a), c V Fiat, Shamir 86: 3 round interactive protocol => 1 round argument Pointcheval, Stern 96: secure in the random oracle model. Goldwasser, Kalai 03: insecure for arguments with any real hash functions. 15
16 P a b (public coins) c V => P a, b=h(a), c V Fiat, Shamir 86: 3 round interactive protocol => 1 round argument Pointcheval, Stern 96: secure in the random oracle model. Goldwasser, Kalai 03: insecure for arguments with any real hash functions. A very popular paradigm, hard to argue security with a concrete property. 16
17 P a b (public coins) c V => P a, b=h(a), c V Fiat, Shamir 86: 3 round interactive protocol => 1 round argument Pointcheval, Stern 96: secure in the random oracle model. Goldwasser, Kalai 03: insecure for arguments with any real hash functions. A very popular paradigm, hard to argue security with a concrete property. Kalai, Rothblum, Rothblum 17: io + more => Fiat-Shamir for proofs 17
18 This talk: 1. Correlation intractability => Fiat Shamir for proofs 2. Show that for T = Enc k (m) H T (x) = Dec x (T) is correlation intractable if (Enc, Dec) is exp. KDM secure. 3. More about correlation intractability 18
19 Part I: What is correlation intractability? 19
20 Correlation Intractability infeasibility of finding sparse input-output relations --- Canetti, Goldreich, Halevi
21 Sparse Relations For every input (x), the fraction of outputs (y) in the relation is negligible 21
22 Sparse Relations For every input (x), the fraction of outputs (y) in the relation is negligible Correlation intractability [Canetti, Goldreich, Halevi 98] 22
23 Sparse Relations For every input (x), the fraction of outputs (y) in the relation is negligible Correlation intractability [Canetti, Goldreich, Halevi 98] For all sparse relations R: 23
24 Sparse Relations For every input (x), the fraction of outputs (y) in the relation is negligible Correlation intractability [Canetti, Goldreich, Halevi 98] For all sparse relations R: Adversary h Challenger 24
25 Sparse Relations For every input (x), the fraction of outputs (y) in the relation is negligible Correlation intractability [Canetti, Goldreich, Halevi 98] For all sparse relations R: Adversary h Challenger x, (as a result, y=h(x)) Adversary wins if R(x, y)=1 25
26 Correlation intractability => Fiat-Shamir for proofs [ Hada, Tanaka 99; Dwork et al. 99 ] 26
27 An interactive proof system A a B b (public coins) c => A a, b=h(a), c B Fiat, Shamir 86: 3 round proof system => 1 round argument 27
28 An interactive proof system A a B b (public coins) c => A a, b=h(a), c B Fiat, Shamir 86: 3 round proof system => 1 round argument Fiat, Shamir relation: (the instance x is part of a or c) R(a, b)=1 if c s.t. x L and Verifier(x, a, b, c) accepts 28
29 An interactive proof system A a B b (public coins) c => A a, b=h(a), c B Fiat, Shamir 86: 3 round proof system => 1 round argument [ Bitansky et al. 13 ] for proof systems, impossible from black-box reductions to falsifiable assumptions. 29
30 More quick facts of correlation intractability Impossible when key/seed is short [ Canetti, Goldreich, Halevi 98 ]. 30
31 More quick facts of correlation intractability Impossible when key/seed is short [ Canetti, Goldreich, Halevi 98 ]. Our goal: capture as many sparse relations as possible, including the relations that cover Fiat-Shamir for proofs. 31
32 Part 2: How to construct correlation intractable functions? 32
33 Existing constructions: 33
34 Existing constructions: 34
35 Existing constructions: 35
36 36
37 37
38 38
39 39
40 40
41 41
42 42
43 43
44 Construction 44
45 45
46 46
47 g ax+b 47
48 [xa+b] 48
49 Analysis 49
50 50
51 51
52 52
53 53
54 54
55 55
56 56
57 57
58 58
59 59
60 60
61 Part 3: More Correlation intractability & Bitcoin 61
62 H(???...?)= XYZ3d83h 62
63 H(???...?)= XYZ3d83h Quantitative correlation intractability: For all relations of density d, all adversaries running in time T succeed with probability f(d, T). 63
64 Quantitative correlation intractability. Multiple-input-output relations. Future directions 64
65
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