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Reliable communication via semilattice properties of partial knowledge

• A. Pagourtzis 1
• G. Panagiotakos 2
• D. Sakavalas 1

1 School of Electrical and Computer Engineering

National Technical University of Athens

2 School of Informatics

University of Edinburgh

21st International Symposium on Fundamentals of Computation Theory Bordeaux, France, September 13, 2017

Distributed Computing in an unreliable environment

– Several interacting entities (players/agents) that cooperate to achieve a common goal without central coordination.

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 2 / 19

Distributed Computing in an unreliable environment

– Several interacting entities (players/agents) that cooperate to achieve a common goal without central coordination. – Players arranged in a communication network G.

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 2 / 19

Distributed Computing in an unreliable environment

– Several interacting entities (players/agents) that cooperate to achieve a common goal without central coordination. – Players arranged in a communication network G. – Adversarial Behavior: Corrupted players controlled by a central active (Byzantine) adversary.

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 2 / 19

Distributed Computing in an unreliable environment

– Several interacting entities (players/agents) that cooperate to achieve a common goal without central coordination. – Players arranged in a communication network G. – Adversarial Behavior: Corrupted players controlled by a central active (Byzantine) adversary. – Achieve goal despite the presence of corruptions.

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 2 / 19

Reliable Communication

– Reliable Message Transmission (RMT) problem: Correct delivery of message x from Sender S to Receiver R, despite the existence of corrupted players.

S R

Incomplete Network

G = (V, E)

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 3 / 19

Reliable Communication

– Reliable Message Transmission (RMT) problem: Correct delivery of message x from Sender S to Receiver R, despite the existence of corrupted players. Sender’s input: x

x

S R

x x

relay

nodes G = (V, E)

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 3 / 19

Reliable Communication

– Reliable Message Transmission (RMT) problem: Correct delivery of message x from Sender S to Receiver R, despite the existence of corrupted players. (Sender’s input: x, Receiver’s output (decision): x)

S R

x x x x

G = (V, E)

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 3 / 19

Reliable Communication

– Reliable Message Transmission (RMT) problem: Correct delivery of message x from Sender S to Receiver R, despite the existence of corrupted players. (Sender’s input: x, Receiver’s output (decision): x)

? ?

S R

x y y

G = (V, E)

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 3 / 19

Reliable Communication

– Reliable Message Transmission (RMT) problem: Correct delivery of message x from Sender S to Receiver R, despite the existence of corrupted players. (Sender’s input: x, Receiver’s output (decision): x)

S R

x

??

(simulation of a reliable channel)

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 3 / 19

Reliable Communication

– Reliable Message Transmission (RMT) problem: Correct delivery of message x from Sender S to Receiver R, despite the existence of corrupted players. (Sender’s input: x, Receiver’s output (decision): x)

S R

x

??

(simulation of a reliable channel)

– Main result: Exact characterization of instances where RMT is feasible (impossibility condition, matching algorithm)

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 3 / 19

The Adversary

Corruption sets – t-Global [Lamport, Shostak, Pease, ’82]: At most t corruptions.

At most t corruptions

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 4 / 19

The Adversary

Corruption sets – t-Global [Lamport, Shostak, Pease, ’82]: At most t corruptions. – t-Local [Koo, ’04]: At most t corruptions in each neighborhood

· · · · · ·

{

At most t corruptions

{

At most t corruptions

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 4 / 19

The Adversary

Corruption sets – t-Global [Lamport, Shostak, Pease, ’82]: At most t corruptions. – t-Local [Koo, ’04]: At most t corruptions in each neighborhood – General Adversary [Hirt, Maurer, ’97]: Defined by the monotone family of all possible corruption sets Z ⊆ 2V (adversary structure).

Z3 Z1 Z2 Z = {Z1, Z2, Z3, Z4} Z4

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 4 / 19

Initial knowledge of players

Partial knowledge model [Pagourtzis, Panagiotakos, Sakavalas, ’14] – Topology knowledge: Player u knows subgraph γ(u) = (Vu, Eu). u γ(u)

w γ(w)

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 5 / 19

Initial knowledge of players

Partial knowledge model [Pagourtzis, Panagiotakos, Sakavalas, ’14] – Topology knowledge: Player u knows subgraph γ(u) = (Vu, Eu). For set S ⊆ V , γ(S) = ( ∪

u∈S

Vu, ∪

u∈S

Eu). u γ(u)

w γ(w) γ({u, w}) w u

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 5 / 19

Initial knowledge of players

Partial knowledge model [Pagourtzis, Panagiotakos, Sakavalas, ’14] – Topology knowledge: Player u knows subgraph γ(u) = (Vu, Eu). For set S ⊆ V , γ(S) = ( ∪

u∈S

Vu, ∪

u∈S

Eu). u γ(u)

w γ(w) γ({u, w}) w u

– Knowledge of the adversary structure: Each player u knows only the local adversary structure Zu = {S ∩ Vu : S ∈ Z} (also denoted as ZVu).

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 5 / 19

Initial knowledge of players

Partial knowledge model [Pagourtzis, Panagiotakos, Sakavalas, ’14] – Topology knowledge: Player u knows subgraph γ(u) = (Vu, Eu). For set S ⊆ V , γ(S) = ( ∪

u∈S

Vu, ∪

u∈S

Eu). u γ(u)

w γ(w) γ({u, w}) w u

– Knowledge of the adversary structure: Each player u knows only the local adversary structure Zu = {S ∩ Vu : S ∈ Z} (also denoted as ZVu).

γ(v) v Z1 Z2 Z3

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 5 / 19

Initial knowledge of players

Partial knowledge model [Pagourtzis, Panagiotakos, Sakavalas, ’14] – Topology knowledge: Player u knows subgraph γ(u) = (Vu, Eu). For set S ⊆ V , γ(S) = ( ∪

u∈S

Vu, ∪

u∈S

Eu). u γ(u)

w γ(w) γ({u, w}) w u

– Knowledge of the adversary structure: Each player u knows only the local adversary structure Zu = {S ∩ Vu : S ∈ Z} (also denoted as ZVu).

γ(v) v Z1 Z2

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 5 / 19

The Model

Adversary – Byzantine. – General. – Unbounded.

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 6 / 19

The Model

Adversary – Byzantine. – General. – Unbounded. Network – Arbitrary topology (aka incomplete). – Synchronous. – Authenticated channels (no tampering, known sender id).

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 6 / 19

The Model

Adversary – Byzantine. – General. – Unbounded. Network – Arbitrary topology (aka incomplete). – Synchronous. – Authenticated channels (no tampering, known sender id). Initial knowledge – Partial knowledge over topology and adversary. Safe RMT algorithms [Pelc, Peleg, ’05] – Never make the receiver output (decide on) an incorrect value.

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 6 / 19

Intuition behind RMT protocols

– Known Topology: R decides on x upon receiving x from a set of S ⇝ R paths not fully “covered” by a corruptible set.

. . . R S

x x x

G

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 7 / 19

Intuition behind RMT protocols

– Known Topology: R decides on x upon receiving x from a set of S ⇝ R paths not fully “covered” by a corruptible set.

G

. . . R S

No adversary cover exists x x x

T

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 7 / 19

Intuition behind RMT protocols

– Known Topology: R decides on x upon receiving x from a set of S ⇝ R paths not fully “covered” by a corruptible set. – Partial knowledge: Node v decides on x upon receipt from a set of paths in γ(v) not “covered” by a corruptible set.

γ(v)

. . . v

w u G

w decided on x u decided on x

S

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 7 / 19

Intuition behind RMT protocols

– Known Topology: R decides on x upon receiving x from a set of S ⇝ R paths not fully “covered” by a corruptible set. – Partial knowledge: Node v decides on x upon receipt from a set of paths in γ(v) not “covered” by a corruptible set.

γ(v)

. . . v

w decided on x u decided on x

w u G

No adversary cover exists

S

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 7 / 19

Intuition behind RMT protocols

– Known Topology: R decides on x upon receiving x from a set of S ⇝ R paths not fully “covered” by a corruptible set. – Partial knowledge: Node v decides on x upon receipt from a set of paths in γ(v) not “covered” by a corruptible set.

γ(v)

. . . v

w decided on x u decided on x

w u G

No adversary cover exists

S

Algorithm (GPPA) tight for local knowledge [PPS14]!

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 7 / 19

Exchange of Knowledge

– GPPA not generally tight. – Knowledge exchange helps in the general case.

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 8 / 19

Exchange of Knowledge

– GPPA not generally tight. – Knowledge exchange helps in the general case.

. . . . . .

Paths P contain an adversary cover in γ(v)

v γ(v)

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 8 / 19

Exchange of Knowledge

– GPPA not generally tight. – Knowledge exchange helps in the general case.

. . . . . .

w γ({v, w})

Knowledge exchange Paths P contain an adversary cover in γ(v)

v γ(v)

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 8 / 19

Exchange of Knowledge

– GPPA not generally tight. – Knowledge exchange helps in the general case.

. . .

Paths P ′ do not contain an adversary cover in

. . .

w γ({v, w})

Knowledge exchange Paths P contain an adversary cover in γ(v)

v γ(v)

γ({v, w})

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 8 / 19

Exchange of Knowledge

– GPPA not generally tight. – Knowledge exchange helps in the general case.

. . .

Paths P ′ do not contain an adversary cover in

. . .

w γ({v, w})

Knowledge exchange Paths P contain an adversary cover in γ(v)

v γ(v)

γ({v, w})

Knowledge Exchange between v, w – Joining topology knowledge: trivially γ({v, w}) – Joining local adversary structures Zv, Zw?

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 8 / 19

Joining Local Adversary Structures

ZA = {Z ∩ A | Z ∈ Z}. For ZA, ZB, define (worst case) joint structure ZA ⊕ ZB: . . .

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 9 / 19

Joining Local Adversary Structures

ZA = {Z ∩ A | Z ∈ Z}. For ZA, ZB, define (worst case) joint structure ZA ⊕ ZB:

A B Z1 Z2

– ZA ⊕ ZB should contain Z1 ∪ Z2. . . .

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 9 / 19

Joining Local Adversary Structures

ZA = {Z ∩ A | Z ∈ Z}. For ZA, ZB, define (worst case) joint structure ZA ⊕ ZB:

A B Z1 Z2 Z3 Z4

– ZA ⊕ ZB should contain Z1 ∪ Z2. – ZA ⊕ ZB should contain Z3 ∪ Z4. . . .

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 9 / 19

Joining Local Adversary Structures

ZA = {Z ∩ A | Z ∈ Z}. For ZA, ZB, define (worst case) joint structure ZA ⊕ ZB:

A B Z1 Z2 Z3 Z4 Z5 Z6

– ZA ⊕ ZB should contain Z1 ∪ Z2. – ZA ⊕ ZB should contain Z3 ∪ Z4. – ZA ⊕ ZB should not contain Z5 ∪ Z6. . . .

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 9 / 19

Joining Local Adversary Structures

ZA = {Z ∩ A | Z ∈ Z}. For ZA, ZB, define (worst case) joint structure ZA ⊕ ZB:

A B Z1 Z2 Z3 Z4 Z5 Z6

– ZA ⊕ ZB should contain Z1 ∪ Z2. – ZA ⊕ ZB should contain Z3 ∪ Z4. – ZA ⊕ ZB should not contain Z5 ∪ Z6. .

Join Operation ⊕

. . ZA ⊕ ZB = {Z1 ∪ Z2|(Z1 ∈ ZA) ∧ (Z2 ∈ ZB) ∧ (Z1 ∩ B = Z2 ∩ A)}

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 9 / 19

Joining Local Adversary Structures

ZA = {Z ∩ A | Z ∈ Z}. For ZA, ZB, define (worst case) joint structure ZA ⊕ ZB:

A B Z1 Z2 Z3 Z4 Z5 Z6

– ZA ⊕ ZB should contain Z1 ∪ Z2. – ZA ⊕ ZB should contain Z3 ∪ Z4. – ZA ⊕ ZB should not contain Z5 ∪ Z6. .

Join Operation ⊕

. . ZA ⊕ ZB = {Z1 ∪ Z2|(Z1 ∈ ZA) ∧ (Z2 ∈ ZB) ∧ (Z1 ∩ B = Z2 ∩ A)} Definition extends to different structures: ZA ⊕ Z′B (false structure report by adversary).

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 9 / 19

Semilattice structure of partial knowledge

.

Theorem

. . Let T = {ZA | Z ⊆ 2V , A ⊆ V } the space of all possible ZA. ⟨T, ⊕⟩ is a semilattice. . . .

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 10 / 19

Semilattice structure of partial knowledge

.

Theorem

. . Let T = {ZA | Z ⊆ 2V , A ⊆ V } the space of all possible ZA. ⟨T, ⊕⟩ is a semilattice.

• Proof. Operation ⊕ is commutative, associative, idempotent.

. . .

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 10 / 19

Semilattice structure of partial knowledge

.

Theorem

. . Let T = {ZA | Z ⊆ 2V , A ⊆ V } the space of all possible ZA. ⟨T, ⊕⟩ is a semilattice.

• Proof. Operation ⊕ is commutative, associative, idempotent.

.

Semilattices facts

. .

• 1. ⊕ induces a partial order “ ≽ ” on T by x ≽ y ⇔ x = x ⊕ y.
• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 10 / 19

Semilattice structure of partial knowledge

.

Theorem

. . Let T = {ZA | Z ⊆ 2V , A ⊆ V } the space of all possible ZA. ⟨T, ⊕⟩ is a semilattice.

• Proof. Operation ⊕ is commutative, associative, idempotent.

.

Semilattices facts

. .

• 1. ⊕ induces a partial order “ ≽ ” on T by x ≽ y ⇔ x = x ⊕ y.
• 2. Every nonempty finite subset of T has a supremum w.r.t. ≽.
• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 10 / 19

Semilattice structure of partial knowledge

.

Theorem

. . Let T = {ZA | Z ⊆ 2V , A ⊆ V } the space of all possible ZA. ⟨T, ⊕⟩ is a semilattice.

• Proof. Operation ⊕ is commutative, associative, idempotent.

.

Semilattices facts

. .

• 1. ⊕ induces a partial order “ ≽ ” on T by x ≽ y ⇔ x = x ⊕ y.
• 2. Every nonempty finite subset of T has a supremum w.r.t. ≽.
• 3. sup{x, y} = x ⊕ y (join).
• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 10 / 19

Semilattice structure of partial knowledgetle

.

Theorem (Induced partial order)

. . Operation ⊕ induces partial order “ ≽ ” on T: ZA ≽ Z′B ⇔ (A ⊇ B) ∧ ( (ZA)B ⊆ Z′B)

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 11 / 19

Semilattice structure of partial knowledgetle

.

Theorem (Induced partial order)

. . Operation ⊕ induces partial order “ ≽ ” on T: ZA ≽ Z′B ⇔ (A ⊇ B) ∧ ( (ZA)B ⊆ Z′B)

A B ZA

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 11 / 19

Semilattice structure of partial knowledgetle

.

Theorem (Induced partial order)

. . Operation ⊕ induces partial order “ ≽ ” on T: ZA ≽ Z′B ⇔ (A ⊇ B) ∧ ( (ZA)B ⊆ Z′B)

A B ZB

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 11 / 19

Semilattice structure of partial knowledgetle

.

Theorem (Induced partial order)

. . Operation ⊕ induces partial order “ ≽ ” on T: ZA ≽ Z′B ⇔ (A ⊇ B) ∧ ( (ZA)B ⊆ Z′B)

A B ZB

• Proof. By (1) and definition of ⊕.
• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 11 / 19

Maximality of ZA ⊕ ZB

A B Z1 Z2 Z A B Z′ Z3

. . .

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 12 / 19

Maximality of ZA ⊕ ZB

A B Z1 ZA A B Z1 Z′A

. . .

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 12 / 19

Maximality of ZA ⊕ ZB

A B Z2 ZB A B Z2 ZB

. . .

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 12 / 19

Maximality of ZA ⊕ ZB

A B Z3 ZA ⊕ ZB A B Z3 Z′A ⊕ Z′B

. . .

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 12 / 19

Maximality of ZA ⊕ ZB

A B Z3 ZA ⊕ ZB A B Z3 Z′A ⊕ Z′B

– ZA ⊕ ZB: worst possible case compatible with local knowledge? . . .

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 12 / 19

Maximality of ZA ⊕ ZB

A B Z3 ZA ⊕ ZB A B Z3 Z′A ⊕ Z′B

– ZA ⊕ ZB: worst possible case compatible with local knowledge? – Are there corruptible subsets of A ∪ B not included in ZA ⊕ ZB? . . .

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 12 / 19

Maximality of ZA ⊕ ZB

A B Z3 ZA ⊕ ZB A B Z3 Z′A ⊕ Z′B

– ZA ⊕ ZB: worst possible case compatible with local knowledge? – Are there corruptible subsets of A ∪ B not included in ZA ⊕ ZB? .

Theorem

. . For structure Z and A, B ⊆ V , it holds that Z(A∪B) ⊆ ZA ⊕ ZB.

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 12 / 19

Maximality of ZA ⊕ ZB

A B Z3 ZA ⊕ ZB A B Z3 Z′A ⊕ Z′B

– ZA ⊕ ZB: worst possible case compatible with local knowledge? – Are there corruptible subsets of A ∪ B not included in ZA ⊕ ZB? .

Theorem

. . For structure Z and A, B ⊆ V , it holds that Z(A∪B) ⊆ ZA ⊕ ZB. Proof. Z(A∪B) ≽ ZA, ZB ZA ⊕ ZB = sup{ZA, ZB} } ⇒ Z(A∪B) ≽ ZA ⊕ ZB ⇒ . . .

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 12 / 19

RMT cut

Joint Adversary Structure: Maximum possible adversary structure w.r.t the initial knowledge of players in B. ZB = ⊕

u∈B

ZVu . . . .

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 13 / 19

RMT cut

Joint Adversary Structure: Maximum possible adversary structure w.r.t the initial knowledge of players in B. ZB = ⊕

u∈B

ZVu .

RMT cut

. . Node cut C = T ∪ H, cut of G, disconnecting S ∈ A from R ∈ B s.t. T ∈ Z and H ∩ γ(B) ∈ ZB. (B: connected component)

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 13 / 19

RMT cut

Joint Adversary Structure: Maximum possible adversary structure w.r.t the initial knowledge of players in B. ZB = ⊕

u∈B

ZVu .

RMT cut

. . Node cut C = T ∪ H, cut of G, disconnecting S ∈ A from R ∈ B s.t. T ∈ Z and H ∩ γ(B) ∈ ZB. (B: connected component)

A B

T H

S R

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 13 / 19

RMT cut

Joint Adversary Structure: Maximum possible adversary structure w.r.t the initial knowledge of players in B. ZB = ⊕

u∈B

ZVu .

RMT cut

. . Node cut C = T ∪ H, cut of G, disconnecting S ∈ A from R ∈ B s.t. T ∈ Z and H ∩ γ(B) ∈ ZB. (B: connected component)

A B

T ∈ Z

S R γ(B)

H′ ∈ ZB

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 13 / 19

RMT cut

Joint Adversary Structure: Maximum possible adversary structure w.r.t the initial knowledge of players in B. ZB = ⊕

u∈B

ZVu .

RMT cut

. . Node cut C = T ∪ H, cut of G, disconnecting S ∈ A from R ∈ B s.t. T ∈ Z and H ∩ γ(B) ∈ ZB. (B: connected component)

A B

T ∈ Z

S R γ(B)

H′ ∈ ZB

– T is corruptible. – H “looks” corruptible to B.

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 13 / 19

RMT Impossibility

.

Theorem (Necessary condition for safe RMT)

. . If an RMT-cut exists for instance (G, Z, S, R) then no safe algorithm A can achieve RMT in (G, Z, S, R).

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 14 / 19

RMT Impossibility

.

Theorem (Necessary condition for safe RMT)

. . If an RMT-cut exists for instance (G, Z, S, R) then no safe algorithm A can achieve RMT in (G, Z, S, R). * Safe Algorithm [Pelc, Peleg ’05]: Either R is sure for the sender’s value or does not decide at all.

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 14 / 19

RMT Impossibility

.

Theorem (Necessary condition for safe RMT)

. . If an RMT-cut exists for instance (G, Z, S, R) then no safe algorithm A can achieve RMT in (G, Z, S, R). * Safe Algorithm [Pelc, Peleg ’05]: Either R is sure for the sender’s value or does not decide at all. (roughly non-safe makes assumptions that might not hold.)

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 14 / 19

Proof Sketch

(Conflicting indistinguishable executions)

Assume that a safe algorithm A achieves RMT in (G, Z, S, R) with RMT cut C = T ∪ H. What about (G, ZB, S, R)?

S

T A B H

R

(G, Z, S, R) Run r0

S

T A B H

R

(G, ZB, S, R) Run r0

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 15 / 19

Proof Sketch

(Conflicting indistinguishable executions)

Assume that a safe algorithm A achieves RMT in (G, Z, S, R) with RMT cut C = T ∪ H. What about (G, ZB, S, R)?

S

T A B H (G, Z, S, R)

R

(x = 0) Run r0 decision(R) = 0

S

T A B H (G, ZB, S, R)

R

(x = 1) Run r1 decision(R) =??

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 15 / 19

Proof Sketch

(Conflicting indistinguishable executions)

Assume that a safe algorithm A achieves RMT in (G, Z, S, R) with RMT cut C = T ∪ H. What about (G, ZB, S, R)?

S

T A B H (G, Z, S, R)

R

x = 1 x = 0 (x = 0) Run r0 decision(R) = 0

S

T A B H (G, ZB, S, R)

R

x = 1 x = 0 (x = 1) Run r1 decision(R) =??

Corrupted players of ri act as honest in r1−i.

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 15 / 19

Proof Sketch

(Conflicting indistinguishable executions)

Assume that a safe algorithm A achieves RMT in (G, Z, S, R) with RMT cut C = T ∪ H. What about (G, ZB, S, R)?

S

T A B H (G, Z, S, R)

R

x = 1 x = 0 (x = 0) Run r0 decision(R) = 0

S

T A B H (G, ZB, S, R)

R

x = 1 x = 0 (x = 1) Run r1 decision(R) =??

Corrupted players of ri act as honest in r1−i. – Runs r0, r1, indistinguishable to the set of nodes B (same joint knowledge and joint view).

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 15 / 19

Proof Sketch

(Conflicting indistinguishable executions)

Assume that a safe algorithm A achieves RMT in (G, Z, S, R) with RMT cut C = T ∪ H. What about (G, ZB, S, R)?

S

T A B H (G, Z, S, R)

R

x = 1 x = 0 (x = 0) Run r0 decision(R) = 0

S

T A B H (G, ZB, S, R)

R

x = 1 x = 0 (x = 1) Run r1 decision(R) = 0

Corrupted players of ri act as honest in r1−i. – Runs r0, r1, indistinguishable to the set of nodes B (same joint knowledge and joint view). – R decides on the same value 0 in both runs, thus A is not safe.

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 15 / 19

RMT- Partial Knowledge Algorithm

.

RMT-PKA Outline

. . Propagation Phase – Dealer’s value is propagated throughout the graph.

x S R

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 16 / 19

RMT- Partial Knowledge Algorithm

.

RMT-PKA Outline

. . Propagation Phase – Dealer’s value is propagated throughout the graph. – Each player propagates its initial knowledge (γ(v), Zv).

x S R

(v, γ(v), Zv)

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 16 / 19

RMT- Partial Knowledge Algorithm

.

RMT-PKA Outline

. . Propagation Phase – Dealer’s value is propagated throughout the graph. – Each player propagates its initial knowledge (γ(v), Zv). Decision Phase – Identifies a “non-contradicting” set of messages M. – Creates subgraph GM implied by messages M.

S R

GM

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 16 / 19

RMT- Partial Knowledge Algorithm

.

RMT-PKA Outline

. . Propagation Phase – Dealer’s value is propagated throughout the graph. – Each player propagates its initial knowledge (γ(v), Zv). Decision Phase – Identifies a “non-contradicting” set of messages M. – Creates subgraph GM implied by messages M. – Decides on value propagated by M if GM does not have an adversary cover. (C ∩ γ(B) ∈ ZB)

S R

A B

γ(v)

C

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 16 / 19

Optimal Resilience of RMT-PKA

.

Theorem (Safety)

. . R will never decide on an incorrect value. . . .

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 17 / 19

Optimal Resilience of RMT-PKA

.

Theorem (Safety)

. . R will never decide on an incorrect value. .

Theorem (Sufficiency of RMT-cut condition)

. . RMT-PKA achieves RMT whenever an RMT-cut does not exist.

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 17 / 19

Optimal Resilience of RMT-PKA

.

Theorem (Safety)

. . R will never decide on an incorrect value. .

Theorem (Sufficiency of RMT-cut condition)

. . RMT-PKA achieves RMT whenever an RMT-cut does not exist. Non-existence of an RMT-cut is a necessary and sufficient condition for achieving RMT

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 17 / 19

Open Questions

– Efficiency study for RMT in the partial knowledge model. (Efficient algorithm known only for the t-local model under local knowledge.)

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 18 / 19

Open Questions

– Efficiency study for RMT in the partial knowledge model. (Efficient algorithm known only for the t-local model under local knowledge.) – Resilience measures and approximation. (Existence of an RMT cut is NP-hard to check.)

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 18 / 19

Open Questions

– Efficiency study for RMT in the partial knowledge model. (Efficient algorithm known only for the t-local model under local knowledge.) – Resilience measures and approximation. (Existence of an RMT cut is NP-hard to check.) – Privacy requirements in partial knowledge models (SMT).

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 18 / 19

Thank you!

• A. Pagourtzis, G. Panagiotakos, D. Sakavalas

RMT under Partial Knowledge 19 / 19