Abstract Storage Devices
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1 Abstract Storage Devices Robert König Ueli Maurer Stefano Tessaro SOFSEM 2009 January 27, 2009
2 Outline 1. Motivation: Storage Devices 2. Abstract Storage Devices (ASD s) 3. Reducibility 4. Factoring ASD s 5. Future Directions
3 Outline 1. Motivation: Storage Devices 2. Abstract Storage Devices (ASD s) 3. Reducibility 4. Factoring ASD s 5. Future Directions
4 Storage Devices
5 Storage Devices Multiple retrieval operations + partial information
6 Storage Devices
7 Storage Devices Motivation Physical laws (e.g. quantum state) Efficiency constraints
8 Storage Devices Cryptographic Applications Information leakage [KB07] Memory-bounded adversaries Motivation Physical laws (e.g. quantum state) Efficiency constraints
9 Outline 1. Motivation: Storage Devices 2. Abstract Storage Devices (ASD s) 3. Reducibility 4. Factoring ASD s 5. Future Directions
10 Deterministic Storage Device This work. We consider deterministic devices.
11 Deterministic Storage Device This work. We consider deterministic devices. Motivation Natural examples Interesting phenomena Combinatorial characterization
12 Deterministic Storage Device This work. We consider deterministic devices. Motivation Natural examples Interesting phenomena Combinatorial characterization Observation. Output labeling irrelevant
13 Abstract Storage Devices (ASD s) Abstract Storage Device (ASD). Ordered pair D = (S, Π) S: state space Π: partition set, i.e. π Π have form { π = B 1,..., B l ( i j : B i B j = ) l i=1 } B i = S.
14 Abstract Storage Devices (ASD s) Abstract Storage Device (ASD). Ordered pair D = (S, Π) S: state space Π: partition set, i.e. π Π have form { π = B 1,..., B l ( i j : B i B j = ) l i=1 } B i = S. π 1 π 2 π 3
15 ASD s Write and Read Write operation Retrieval operation
16 ASD s Write and Read Write operation Retrieval operation
17 ASD s Write and Read Write operation Retrieval operation
18 ASD s Write and Read Write operation Retrieval operation
19 ASD s Write and Read Write operation Retrieval operation
20 ASD s Write and Read Write operation Retrieval operation
21 ASD s Write and Read Write operation Retrieval operation
22 ASD s Write and Read Write operation Retrieval operation In this talk: s s : π : s π s
23 ASD s Examples
24 ASD s Examples
25 ASD s Examples Perfect Device C s : S := {0,..., s 1} Π := {id}, with id := {{0},..., {s 1}}
26 ASD s Examples
27 ASD s Examples Projective Device P n : S := {0, 1} n Π := {π 1,..., π n }, with π i := {{x : x i = 0}, {x : x i = 1}}
28 ASD s Examples
29 ASD s Examples Linear Device L n : S := {0, 1} n, Π := {π a : a {0, 1} n } π a := {{x : a, x = 0}, {x : a, x = 1}}
30 Composing ASD s Direct Products Direct product D D S(D D ) := S(D) S(D ); Π(D D ) := {π π π Π(D), π Π(D )}, where π π := {B B B π, B π }
31 Composing ASD s Sequence Device k-sequence Device D (k) S(D (k) ) := S(D); Π(D (k) ) := {π 1 π k π 1,..., π k Π(D)}, where π π := {B B B π, B π }.
32 Outline 1. Motivation: Storage Devices 2. Abstract Storage Devices (ASD s) 3. Reducibility 4. Factoring ASD s 5. Future Directions
33 ASD Reducibility Question. Is ASD D stronger than D? Can we implement D using D?
34 ASD Reducibility Question. Is ASD D stronger than D? Can we implement D using D? s, g: This work: zero-error reductions
35 ASD Reducibility Definition Reduction D D. Ordered pair (φ, α) with φ : S(D) S(D ) and α : Π(D) Π(D ) such that π Π(D) : α(π) φ refines π
36 ASD Reducibility Definition Reduction s α(π) φ D s D. Ordered φ(s) pair α(π) (φ, φ(s α) ) with φ : S(D) S(D ) and α : Π(D) Π(D ) such that π Π(D) : α(π) φ refines π
37 ASD Reducibility Definition Reduction s α(π) φ D s D. Ordered φ(s) pair α(π) (φ, φ(s α) ) with φ : S(D) S(D ) and α : Π(D) Π(D ) such that π Π(D) : α(π) φ refines π Notation. D D : (φ, α) reduction D D D D : D D D D.
38 ASD Reducibility Example L 2 D
39 ASD Reducibility Example
40 ASD Reducibility Example
41 ASD Reducibility Example π 1
42 ASD Reducibility Example π 1 φ π 1
43 ASD Reducibility Example π 1 φ refines π 01 π 1
44 ASD Reducibility Example π 1 φ refines π 01 α(π 01 ) := π 1 π 1
45 ASD Reducibility Example π 2 φ refines π 10 α(π 10 ) := π 2 π 2
46 ASD Reducibility Example π 3 φ refines π 11 α(π 11 ) := π 3 π 3
47 ASD Reducibility Example (φ, α) is valid reduction L 2 D π 3 φ refines π 11 α(π 11 ) := π 3 π 3
48 ASD Reducibility Complexity Question. Complexity of deciding reducibility?
49 ASD Reducibility Complexity Question. Complexity of deciding reducibility? Theorem. ASD Reducibility is N P-complete.
50 ASD Reducibility Complexity Question. Complexity of deciding reducibility? Theorem. ASD Reducibility is N P-complete. Proof Idea Reducibility is in N P (witness = reduction) (Complexity-theoretic) reduction from CLIQUE G = (V, E) construct graph device D = D(G)
51 NP-Completeness Graph Devices Undirected graph G = (V, E), V 4.
52 NP-Completeness Graph Devices Undirected graph G = (V, E), V 4.
53 NP-Completeness Graph Devices Undirected graph G = (V, E), V 4.
54 NP-Completeness Graph Devices Undirected graph G = (V, E), V 4.
55 NP-Completeness Graph Devices Undirected graph G = (V, E), V 4.
56 NP-Completeness Graph Devices Undirected graph G = (V, E), V 4.
57 NP-Completeness Graph Devices Undirected graph G = (V, E), V 4. e = {v, w} E π e := {{u}, {v}, V \ {u, v}}
58 NP-Completeness Graph Devices Undirected graph G = (V, E), V 4. e = {v, w} E π e := {{u}, {v}, V \ {u, v}} Lemma. G contained in G D(G ) D(G)
59 NP-Completeness Graph Devices Undirected graph G = (V, E), V 4. e = {v, w} E π e := {{u}, {v}, V \ {u, v}} Lemma. G contained in G D(G ) D(G) Corollary. G contains k-clique D(K k ) D(G)
60 ASD Reducibility Some Challenges L 2 L 3 L 2? L2 L 5? L 3 L 2 L 2 L 4 L 3? L 5 L 3 L 4 L 4
61 ASD Reducibility Some Challenges L 2 L 3 L 2? L2 L 5? L 3 L 2 L 2 L 4 L 3? L 5 L 3 L 4 L 4
62 ASD Reducibility Some Challenges L 2 L 3 L 2? L2 L 5? Storage LCapacity. 3 L 2 C(D) := max{log L 2 s LC 4 s L D}. 3? L 5 L 3 L 4 L 4
63 ASD Reducibility Some Challenges L 2 L 3 L 2? L2 L 5? Storage LCapacity. 3 L 2 C(D) := max{log L 2 s LC 4 s L D}. 3? L 5 L 3 L 4 L 4 Properties. D D = C(D) C(D ) C(D D ) = C(D) + C(D ) C(L n ) = 1
64 ASD Reducibility Some Challenges C = 3 C = 2 L 2 L 3 L 2? L2 L 5? Storage LCapacity. 3 L 2 C(D) := max{log L 2 s LC 4 s L D}. 3? L 5 L 3 L 4 L 4 Properties. D D = C(D) C(D ) C(D D ) = C(D) + C(D ) C(L n ) = 1
65 ASD Reducibility Some Challenges C = 3 C = 2 L 2 L 3 L 2 L 2 L 5? Storage LCapacity. 3 L 2 C(D) := max{log L 2 s LC 4 s L D}. 3? L 5 L 3 L 4 L 4 Properties. D D = C(D) C(D ) C(D D ) = C(D) + C(D ) C(L n ) = 1
66 ASD Reducibility Some Challenges L 2 L 3 L 2 L 2 L 5? L 3 L 2 L 2 L 4 L 3? L 5 L 3 L 4 L 4
67 ASD Reducibility Some Challenges Imperfectness Index. i(d) := min{k C S(D) D (k) }. L 2 L 3 L 2 L 2 L 5? L 3 L 2 L 2 L 4 L 3? L 5 L 3 L 4 L 4
68 ASD Reducibility Some Challenges Imperfectness Index. i(d) := min{k C S(D) D (k) }. L 2 L 3 L 2 L 2 L 5? L 3 L 2 L 2 L 4 L 3? L 5 Properties. L 3 L 4 L 4 D D = i(d) i(d ) i(d D ) = max{i(d), i(d )} i(l n ) = n
69 ASD Reducibility Some Challenges Imperfectness Index. i(d) := min{k C S(D) D (k) }. L 2 L 3 L 2 L 2 L 5? i = 3 L 3 L 2 L 2 L 4 L 3 i = 4? L 5 Properties. L 3 L 4 L 4 D D = i(d) i(d ) i(d D ) = max{i(d), i(d )} i(l n ) = n
70 ASD Reducibility Some Challenges Imperfectness Index. i(d) := min{k C S(D) D (k) }. L 2 L 3 L 2 L 2 L 5 i = 3 L 3 L 2 L 2 L 4 L 3 i = 4? L 5 Properties. L 3 L 4 L 4 D D = i(d) i(d ) i(d D ) = max{i(d), i(d )} i(l n ) = n
71 ASD Reducibility Some Challenges L 2 L 3 L 2 L 2 L 5 L 3 L 2 L 2 L 4 L 3? L 5 L 3 L 4 L 4
72 Outline 1. Motivation: Storage Devices 2. Abstract Storage Devices (ASD s) 3. Reducibility 4. Factoring ASD s 5. Future Directions
73 ASD Factorizations D 1,..., D l factorization of D if D D 1 D l. D is prime if D D 1 D 2. implies D 1 or D 2 is trivial. (Open) Question. Is the factorization in prime ASD s unique?
74 ASD Factorizations 2 Lemma. D 1 D m and E 1 E n products of binary ASD s with equal total state size. D 1 D m E 1 E n if and only if partition {J 1,..., J m } of {1,..., n} with D i j J i E j for all i = 1,..., m.
75 ASD Factorizations 2 Lemma. D 1 D m and E 1 E n products of binary ASD s with equal total state size. D 1 D m E 1 E n if and only if partition {J 1,..., J m } of {1,..., n} with D i j J i E j for all i = 1,..., m. Theorem. The factorization of an ASD D in terms of binary ASD s is unique.
76 ASD Factorizations 3 L 2 L 3 L 2 L 2 L 5 L 3 L 2 L 2 L 4 L 3? L 5 L 3 L 4 L 4
77 ASD Factorizations 3 L 2 L 3 L 2 L 2 L 5 L 3 L 2 L 2 L 4 L 3? L 5 L 3 L 4 L 4 L 5 L 4 no partition exists
78 ASD Factorizations 3 L 2 L 3 L 2 L 2 L 5 L 3 L 2 L 2 L 4 L 3 L 5 L 3 L 4 L 4 L 5 L 4 no partition exists
79 Outline 1. Motivation: Storage Devices 2. Abstract Storage Devices (ASD s) 3. Reducibility 4. Factoring ASD s 5. Future Directions
80 Future Directions (Some) Open Problems Study general notions of reducibility Framework for probabilistic storage devices Show unique factorization theorem for ASD s / find counterexamples Find new application scenarios
81 Questions?
82 Motivation: Storage Devices Deterministic Storage Devices ASD - Definition ASD - Read and Write ASD - Examples ASD - Direct Products ASD - Sequence Devices Reducibility - Motivation Reducibility - Definitions Reducibility - Example Reducibility - Complexity Reducibility - Complexity - Proof Sketch Reducibility - Order Preserving Factorizations Factorizations Theorem Factorizations Application Future Directions Motivation: Privacy Amplification
83 Application: Privacy Amplification
84 Application: Privacy Amplification
85 Application: Privacy Amplification
86 Application: Privacy Amplification
87 Application: Privacy Amplification Goal. retrieval ops: Z gives no information about K
88 Application: Privacy Amplification Goal. retrieval ops: Z gives no information about K Previous work classical PA [BBR88,BBCM95] quantum PA [KMR05,RK05]...
89 ASD Equivalence Complexity Theorem. ASD Equivalence N P-complete PH collapses to 2nd level.
90 ASD Equivalence Complexity Theorem. ASD Equivalence N P-complete PH collapses to 2nd level. Remarks Similar result as for GI Unlikely to be N P-complete
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