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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 21 st 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. Incomplete Network G = ( V, E ) R S 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 G = ( V, E ) relay nodes x x x R S 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 ) G = ( V, E ) x x x x R S 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 ) G = ( V, E ) y ? ? y x R S 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 ) ?? x R S (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 ) ?? x R S (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 ⊆ 2 V ( adversary structure ). Z 4 Z 3 Z 1 Z 2 Z = { Z 1 , Z 2 , Z 3 , Z 4 } 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 ) = ( V u , E u ). w u γ ( u ) 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 ) = ( V u , E u ). For set S ⊆ V , γ ( S ) = ( E u ). V u , u ∈ S u ∈ S w w u u γ ( u ) γ ( w ) γ ( { u, 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 ) = ( V u , E u ). For set S ⊆ V , γ ( S ) = ( E u ). V u , u ∈ S u ∈ S w w u u γ ( u ) γ ( w ) γ ( { u, w } ) – Knowledge of the adversary structure: Each player u knows only the local adversary structure Z u = { S ∩ V u : S ∈ Z} (also denoted as Z V 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 ) = ( V u , E u ). For set S ⊆ V , γ ( S ) = ( E u ). V u , u ∈ S u ∈ S w w u u γ ( u ) γ ( w ) γ ( { u, w } ) – Knowledge of the adversary structure: Each player u knows only γ ( v ) Z 2 the local adversary structure Z 1 Z u = { S ∩ V u : S ∈ Z} Z 3 v (also denoted as Z V 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 ) = ( V u , E u ). For set S ⊆ V , γ ( S ) = ( E u ). V u , u ∈ S u ∈ S w w u u γ ( u ) γ ( w ) γ ( { u, w } ) – Knowledge of the adversary structure: Each player u knows only γ ( v ) Z 2 the local adversary structure Z 1 Z u = { S ∩ V u : S ∈ Z} v (also denoted as Z V u ). 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. G x S R x . . x . 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 x S R x . . x . T No adversary cover exists 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. G w decided on x w v γ ( v ) S . . . u u decided on x 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. G w decided on x w v γ ( v ) S . . . u u decided on x No adversary cover exists A. Pagourtzis, G. Panagiotakos, D. Sakavalas RMT under Partial Knowledge 7 / 19