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Robust Mixing Strategies for Scalable and Resilient Overlay Networks

In this paper, we explore advanced mixing techniques for structured overlay networks, aimed at enhancing scalability and robustness in peer-to-peer systems. We address challenges posed by adversarial peers through a join-leave model, proposing mechanisms such as the k-cuckoo rule and k-flip & evict rule. Our analysis ensures that honest peers maintain a majority despite polynomial sequences of rejoining adversarial peers, thereby achieving stable region balancing in dynamic environments. This research contributes to the growing body of knowledge on secure and efficient overlay network design.

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Robust Mixing Strategies for Scalable and Resilient Overlay Networks

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  1. Robust Mixing for Structured Overlay Networks Christian Scheideler Institut für Informatik Technische Universität München

  2. 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

  3. Motivation • V: set of peers, U: virtual space • Each v 2 V mapped to regionR(v) ½ U • Family F of functionsf:U ! U • {v,w} edge ,[F(R(v)) Å R(w)] [ [F(R(w)) Å R(v)] = ;

  4. 0 1 f0 f1 R 0 1 Example • Let U=[0,1). • Region selection: [Karger et al. 97]- nodes v 2 V! random points xv2 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

  5. 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

  6. Join-Leave Model • nhonest peers • nadversarial 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

  7. 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

  8. 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

  9. How to satisfy conditions? CAN: map peers to random points in [0,1)

  10. How to satisfy conditions? Group spreading[AS04]: • Map peers to random points in [0,1) • Limit lifetime of peers Too expensive!

  11. How to satisfy conditions? • Rule that works:k-cuckoo rule n honest n adversarial evict k/n-region < 1-1/k Rejoin: leave and join via k-cuckoo rule

  12. Analysis of k-cuckoo rule • k-region: region of sizek/n starting at integer multiple of k/n • R:fixed set of c log nconsec.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.

  13. Analyis of k-cuckoo rule • R:fixed set of c log nconsecutivek-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.

  14. 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.

  15. 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

  16. k-flip&evict rule • Join: as before (k-cuckoo rule) • Leave: choose randomk-region among c log nneighboringk-regions, flip it with random k region n honest n adversarial flip

  17. k-region O(log n)-region k-flip&evict rule Leave: why flip neighboringk-region??? • Anyk-region: O(log n)-region may lose too many peers

  18. k-flip&evict rule Leave: why flip neighboringk-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

  19. k-flip&evict rule Leave: why flip neighboringk-region??? • Focus on region R of c log nk-regions • At most c log nnew 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)

  20. 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.

  21. 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: ???

  22. Questions?

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