Abstract Storage Devices

Abstract Storage Devices Robert König Ueli Maurer Stefano Tessaro SOFSEM 2009 January 27, 2009

Outline 1. Motivation: Storage Devices 2. Abstract Storage Devices (ASD s) 3. Reducibility 4. Factoring ASD s 5. Future Directions

Storage Devices

Storage Devices Multiple retrieval operations + partial information

Storage Devices

Storage Devices Motivation Physical laws (e.g. quantum state) Efficiency constraints

Storage Devices Cryptographic Applications Information leakage [KB07] Memory-bounded adversaries Motivation Physical laws (e.g. quantum state) Efficiency constraints

Deterministic Storage Device This work. We consider deterministic devices.

Deterministic Storage Device This work. We consider deterministic devices. Motivation Natural examples Interesting phenomena Combinatorial characterization

Deterministic Storage Device This work. We consider deterministic devices. Motivation Natural examples Interesting phenomena Combinatorial characterization Observation. Output labeling irrelevant

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.

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

ASD s Write and Read Write operation Retrieval operation

ASD s Write and Read Write operation Retrieval operation In this talk: s s : π : s π s

ASD s Examples

ASD s Examples Perfect Device C s : S := {0,..., s 1} Π := {id}, with id := {{0},..., {s 1}}

ASD s Examples

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}}

ASD s Examples

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}}

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 }

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 }.

ASD Reducibility Question. Is ASD D stronger than D? Can we implement D using D?

ASD Reducibility Question. Is ASD D stronger than D? Can we implement D using D? s, g: This work: zero-error reductions

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

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

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.

ASD Reducibility Example L 2 D

ASD Reducibility Example

ASD Reducibility Example π 1

ASD Reducibility Example π 1 φ π 1

ASD Reducibility Example π 1 φ refines π 01 π 1

ASD Reducibility Example π 1 φ refines π 01 α(π 01 ) := π 1 π 1

ASD Reducibility Example (φ, α) is valid reduction L 2 D π 3 φ refines π 11 α(π 11 ) := π 3 π 3

ASD Reducibility Complexity Question. Complexity of deciding reducibility?

ASD Reducibility Complexity Question. Complexity of deciding reducibility? Theorem. ASD Reducibility is N P-complete.

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)

NP-Completeness Graph Devices Undirected graph G = (V, E), V 4.

NP-Completeness Graph Devices Undirected graph G = (V, E), V 4. e = {v, w} E π e := {{u}, {v}, V \ {u, v}}

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)

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)

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

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

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

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

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

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

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

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

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

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

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

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?

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.

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.

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

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

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

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

Questions?

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

Application: Privacy Amplification

Application: Privacy Amplification Goal. retrieval ops: Z gives no information about K

Application: Privacy Amplification Goal. retrieval ops: Z gives no information about K Previous work classical PA [BBR88,BBCM95] quantum PA [KMR05,RK05]...

ASD Equivalence Complexity Theorem. ASD Equivalence N P-complete PH collapses to 2nd level.

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