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Robust Mixing for Structured Overlay Networks
Christian Scheideler
Institut für Informatik
Technische Universität München
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Motivation
• Peer-to-peer systems have attracted a lot of attention in recent years
• Many scientific peer-to-peer systems use overlay networks based on virtual space
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Motivation
• V: set of peers, U: virtual space• Each v 2 V mapped to region R(v) ½ U• Family F of functions f:U ! U• {v,w} edge , [F(R(v)) Å R(w)] [ [F(R(w)) Å R(v)] = ;
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Example
• Let U=[0,1).
• Region selection: [Karger et al. 97]- nodes v 2 V ! random points xv 2 U- R(v) = [xv, succ(xv)) (regions form partition of U)
• Family F of functions: [Naor & Wieder 03]- f0: x ! x/2- f1: x ! (x+1)/2
0 1R
0 1
f0 f1
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Scalability and Robustness
Scalability:
• Network has (poly-)logarithmic diameter
• Peers have (poly-)logarithmic degree
Robustness:
• Network can handle large fraction of adversarial peers (i.e. honest peers form single connected component)! join-leave attacks
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Join-Leave Model
• n honest peers• n adversarial peers, <1
Operations:• Join(v): peer v joins the system• Leave(v): peer v leaves the system
Goal: maintain scalability and robustness for any sequence of polynomially many adversarial rejoin (leave+join) requests
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More specific goal
• n honest peers, n adversarial peers
• U=[0,1), region selection via Karger et al.( R(v) = [xv, succ(xv)) )
For any interval I ½ [0,1) of size (c log n)/n:
• Balancing condition: (log n) peers in I
• Majority condition: honest peers in majority
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How to satisfy conditions?
Chord: uses cryptographic hash function to map peers to points in [0,1)
• randomly distributes honest peers• does not randomly distribute adversarial peers
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How to satisfy conditions?
CAN: map peers to random points in [0,1)
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How to satisfy conditions?
Group spreading [AS04]:
• Map peers to random points in [0,1)
• Limit lifetime of peers
Too expensive!
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How to satisfy conditions?
• Rule that works: k-cuckoo rule
evict k/n-region
n honest n adversarial
< 1-1/k
Rejoin: leave and join via k-cuckoo rule
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Analysis of k-cuckoo rule
• k-region: region of size k/n starting at integer multiple of k/n
• R: fixed set of c log n consec. k-regions• New node: not yet replaced after joining• >0: small constant
Lemma: R has at most c log n new nodes.
Lemma: Sum of ages of k-regions in R in (1 § ) (c log n)n/k, w.h.p.
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Analyis of k-cuckoo rule
• R: fixed set of c log n consecutive k-regions• T=(/)log3 n• >0: small constant
Lemma: In any time interval of size T, (1§)kT honest nodes and (1§)kT adv. nodes evicted, w.h.p.
Lemma: R has (1§ )(c log n)k old honest and <(1+)(c log n)k old adv. nodes, w.h.p.
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Analysis of k-cuckoo rule
# honest nodes in R: >(1-)(c log n)k
# adversarial nodes in R:<(1+)(c log n)k + (c log n)
Theorem: When using the k-cuckoo rule with <1-1/k, the balancing and majority conditions are satisfied for poly many adversarial rejoin requests, w.h.p.
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Limitation of k-cuckoo rule
• Only works for any sequence of rejoin requests of adversarial peers.
• Does not work for any sequence of rejoin requests.
Example: adversary orders all peers in a region of size O(log n / n) to leave
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k-flip&evict rule
• Join: as before (k-cuckoo rule)
• Leave: choose random k-region among c log n
neighboring k-regions, flip it with random k region
n honest n adversarial
flip
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k-flip&evict rule
Leave: why flip neighboring k-region???
• Any k-region: O(log n)-region may lose too many peers
O(log n)-region
k-region
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k-flip&evict rule
Leave: why flip neighboring k-region???
• k-region of leaving peer: k-regions in O(log n)-region may become too young
• Age distribution:
• O(log n) attempts to replace k-region with k-region of age O(n/log n)
# O(log n)-regions
age
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k-flip&evict rule
Leave: why flip neighboring k-region???
• Focus on region R of c log n k-regions
• At most c log n new nodes in R
• <(1+)c log n nodes left k-regions before they joined R, w.h.p.
• <(1+)c log n nodes left k-regions after they joined R, w.h.p.
• Total age of k-regions > (1-)(c log n)(n/k)
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Analysis of k-flip&evict rule
# honest nodes in R: >(1-)(c log n)k – (1+)(c log n)2
# adversarial nodes in R:<(1+)(c log n)k + (c log n)
Theorem: When using the k-flip&evict rule with <1-3/k, the balancing and majority conditions are satisfied for poly many rejoin requests, w.h.p.
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Conclusion
• Light-weight perturbation rules against join-leave attacks possible
• Recent paper at SPAA 06
• Problems in real world:DoS-attacks, random number generation
• RNG: to appear at OPODIS 06
• DoS: ???
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Questions?