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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