Reliable Broadcase in Unknown Fixed-Identity Networks

Lakshminarayanan Subramanian and Randy H. Katz and Volker Roth and Scott Shenker and Ion Stoica

EECS Department, University of California, Berkeley

Technical Report No. UCB/CSD-04-1358

, 2004

http://www2.eecs.berkeley.edu/Pubs/TechRpts/2004/CSD-04-1358.pdf

In this paper, we formulate a new theoretical problem, namely the reliable broadcast problem in unknown fixed-identity networks. This problem arises in the context of developing decentralized security mechanisms in a specific-class of distributed systems: Consider an undirected graph G connecting n nodes where each node is aware of only its neighbors but not of the entire graph. Additionally, each node has a unique identity and cannot fake its identity to its neighbors. Assume that k among the n nodes act in an adversarial manner and the remaining n-k are good nodes. Under what constraints does there exist a distributed algorithm Gamma that enables every good node v to reliably broadcast a message m(v) to all other good nodes in G? While good nodes follow the algorithm Gamma, an adversary can additionally discard messages, generate spurious messages or collude with other adversaries. In this paper, we prove two results on this problem. First, we provide a distributed algorithm Gamma that can achieve reliable broadcast in an unknown fixed-identity network in the presence of k adversaries if G is 2k + 1 vertex connected. Additionally, a minimum vertex connectivity of 2k + 1 is a necessary condition for achieving reliable broadcast. Next, we study the problem of reliable broadcast in sparse networks (1-connected and 2-connected) in the presence of a single adversary i.e., k = 1. In sparse networks, we show that a single adversary can partition the good nodes into groups such that nodes within a group can reliably broadcast to each other but nodes across groups cannot. For 1-connected and 2-connected graphs, we prove lower bounds on the number of such groups and provide a distributed algorithm to achieve these lower bounds. We also show that in a power-law random graph G(n, alpha), a single adversary can partition at most O(n^1/alpha * (log n)^(5-alpha)/(3-alpha)) good nodes from the remaining set of good nodes. Addressing this problem has practical implications to two real-world problems of paramount importance: (a) developing decentralized security measures to protect Internet routing against adversaries; (b) achieving decentralized public key distribution in static networks. Prior works on Byzantine agreement are not applicable for this problem since they assume that either G is known, or that every pair of nodes can directly communicate, or that nodes use a key distribution infrastructure to sign messages. A solution to our problem can be extended to solve the byzantine agreement problem in unknown fixed-identity networks.

BibTeX citation:

@techreport{Subramanian:CSD-04-1358,
    Author= {Subramanian, Lakshminarayanan and Katz, Randy H. and Roth, Volker and Shenker, Scott and Stoica, Ion},
    Title= {Reliable Broadcase in Unknown Fixed-Identity Networks},
    Year= {2004},
    Url= {http://www2.eecs.berkeley.edu/Pubs/TechRpts/2004/6505.html},
    Number= {UCB/CSD-04-1358},
    Abstract= {In this paper, we formulate a new theoretical problem, namely the reliable broadcast problem in unknown fixed-identity networks. This problem arises in the context of developing decentralized security mechanisms in a specific-class of distributed systems: Consider an undirected graph <i>G</i> connecting <i>n</i> nodes where each node is aware of only its neighbors but not of the entire graph. Additionally, each node has a unique identity and cannot fake its identity to its neighbors. Assume that <i>k</i> among the <i>n</i> nodes act in an adversarial manner and the remaining <i>n</i>-<i>k</i> are good nodes. Under what constraints does there exist a distributed algorithm <i>Gamma</i> that enables every good node <i>v</i> to reliably broadcast a message <i>m</i>(<i>v</i>) to all other good nodes in <i>G</i>? While good nodes follow the algorithm <i>Gamma</i>, an adversary can additionally discard messages, generate spurious messages or collude with other adversaries. <p> In this paper, we prove two results on this problem. First, we provide a distributed algorithm <i>Gamma</i> that can achieve reliable broadcast in an unknown fixed-identity network in the presence of <i>k</i> adversaries if <i>G</i> is <i>2k</i> + 1 vertex connected. Additionally, a minimum vertex connectivity of <i>2k</i> + 1 is a necessary condition for achieving reliable broadcast. Next, we study the problem of reliable broadcast in sparse networks (1-connected and 2-connected) in the presence of a single adversary i.e., <i>k</i> = 1. In sparse networks, we show that a single adversary can partition the good nodes into groups such that nodes within a group can reliably broadcast to each other but nodes across groups cannot. For 1-connected and 2-connected graphs, we prove lower bounds on the number of such groups and provide a distributed algorithm to achieve these lower bounds. We also show that in a power-law random graph <i>G</i>(<i>n</i>, alpha), a single adversary can partition at most <i>O</i>(<i>n</i>^1/alpha * (log <i>n</i>)^(5-alpha)/(3-alpha)) good nodes from the remaining set of good nodes. <p> Addressing this problem has practical implications to two real-world problems of paramount importance: (a) developing decentralized security measures to protect Internet routing against adversaries; (b) achieving decentralized public key distribution in static networks. Prior works on Byzantine agreement are not applicable for this problem since they assume that either <i>G</i> is known, or that every pair of nodes can directly communicate, or that nodes use a key distribution infrastructure to sign messages. A solution to our problem can be extended to solve the byzantine agreement problem in unknown fixed-identity networks.},
}

EndNote citation:

%0 Report
%A Subramanian, Lakshminarayanan 
%A Katz, Randy H. 
%A Roth, Volker 
%A Shenker, Scott 
%A Stoica, Ion 
%T Reliable Broadcase in Unknown Fixed-Identity Networks
%I EECS Department, University of California, Berkeley
%D 2004
%@ UCB/CSD-04-1358
%U http://www2.eecs.berkeley.edu/Pubs/TechRpts/2004/6505.html
%F Subramanian:CSD-04-1358