On Designing Collusion-resistant Routing Schemes for Non-cooperative Wireless Ad Hoc Networks

1. On Designing Collusion-resistant Routing Schemes for Non-cooperative Wireless Ad Hoc Networks Sheng Zhong and Fan Wu Speaker: Fan Wu SUNY at Buffalo Sep.13, 2007

2. Motivation • Wireless ad hoc networks • No infrastructure • Functioning depends on cooperation • Incentives for individual nodes are provided to ensure cooperation [AE03, ERS05, WLW04, ZLL+05, WEW+06], assuming no collusion. • What if nodes collude?

3. State of Art on Incentives in Ad Hoc Networks • Forwarding • Marti et al., MobiCom’00: Watchdog and Pathrater • Buchegger and Le Boudec, MobiHoc’02: Reputation system • Srinivasan et al., INFOCOM’03: generous TIT-FOR-TAT • Buttyan and Hubaux, MobiHoc’00: Nuglets (virtual currency) • Zhong et al., INFOCOM’03: SPRITE, no temper-proof hardware

4. State of Art on Incentives in Ad Hoc Networks (Cont’d) • Routing • Anderegg and Eidenbenz, MobiCom’03: ad hoc-VCG • Eidenbenz et al., IPDPS’05: COMMIT • Wang et al., MobiCom’04: truthful multicast routing • Wang et al., MobiCom’06: OURS, reducing over-payment • Zhong et al., MobiCom’05: integrated approach of game theory and cryptography • Wang and Li, TMC: True Group Strategyproofness

5. A Model of Ad-Hoc Game • A wireless ad hoc network G=(V, E) • V: set of nodes • E: set of edges • Routing game • ci: cost for sending a unit of data • S: source node • D: destination node • V - {S, D}: player node set • ai: node vi ’s action • action profile:

6. A Model of Ad-Hoc Games (Cont’d) • The utility of node vi is: • n: units of data • pi(a): payment for each unit of data • pi’(a): one-time payment • σi(a): • 1: on selected path • 0: not • form the path from S to D

7. Group Strategyproof Equilibrium and Admissibility • Group Strategyproof Equilibrium (GSE): • An action profile a* is a GSE if C V - {S, D}, c, a, nN+ • eithervi C, • or vi C, • Admissible Equilibrium: • Social efficiency: form the lowest-(real)cost path from S to D • nN+ , vi  V - {S, D},

8. Can Admissible GSE Be Achieved? • Answer: No! • General idea of proof: • GSE Strategyproof Equilibrium • We prove it by showing that any Strategyproof Equilibrium a* is not Group Strategyproof, if it is admissible.

9. i0 D S i ci C i0 D S i c’i C Proof: Impossibility of Group Strategyproofness • Lemma: When c is the cost profile, vi C, where S1: cost profile is c S2: cost profile is

10. i0 i0 D D S S i i ci C C c’i Proof: Impossibility of Group Strategyproofness (Cont’d) • Consider vi C, i≠i0 • In S2, since a* is Strategyproof, • In S1, • Consider • So a* is not Group Strategyproof. S1: cost profile is c S2: cost profile is

11. 1 2 2 2 A A B B S S D D E E F F 3 4 6 5 Collusion Beats VCG collude VCG utility: uA=8-3=5 uB=8-3=5 uE=uF=0 u’A=10-4=6 u’B=10-4=6 u’E=u’F=0 uA<u’A uE=uF=u’E=u’F VCG is not Group Strategyproof

12. Strong Nash Equilibrium • Strong Nash Equilibrium (SNE): • An action profile a* is a SNE if C V - {S, D}, c, aC, nN+, there exists a player node vi C such that,

13. Scheme Achieving Admissible SNE • Key idea: discretization of costs • εR+: a very small real number • Every player node’s cost is a multiple of ε. • Claimed cost is also a multiple of ε. • Scheme: • Each player node viclaims its cost ai  R+∪{0} • Choose the lowest-cost path (LCP) from S to D • Tie breaking: lexicographical order

14. Scheme Achieving Strong Nash Equilibrium (Cont’d) • Payments: • For each node vi in the LCP: • Get pi(a) for each unit of data: pi(a) = ai • Receives a one-time payment p’i(a) • For each node vi not in the LCP: • No payment pi(a) • Receives a one-time payment p’i(a)

15. Major Result 1 • Existence of Nash Equilibrium • Construction: • Initialize a* = c • For each if and then a*i← a*i-ε • For each if then a*i← a*i+ε • Repeat until no change to a*

16. 1 1 2 2 A A B B S S D D E E F F 3 3 5 5 Major Result 1 (Cont’d) • Example: • ε=1 • Tie breaking: prefer {A, B} to {E, F} 2 3 2 1 4 {2, 3, 1, 4}

17. Major Result 2 • Lemma: The lowest-cost path is always selected in all Nash Equilibria. • Lemma: For all Nash Equilibrium a* , we have • Theorem: If the above scheme is used, then all Nash Equilibria are admissible Strong Nash Equilibria.

18. Preventing Profit Transfer • GSE and SNE are applicable if profit cannot be transferred between colluding nodes. • But in many practical scenarios, the assumption is not immediately valid. • We provide a method to prevent colluding nodes from transferring profit.

19. ↑10 ↑1 ↑9 A A A 7 7 7 B B B ↓5 ↓2 ↓4 Preventing Profit Transfer • Idea: We make it impossible for colluding nodes to convince each other that they have taken the actions required by the collusion. B takes action a’B instead of aB A takes action a’A instead of aA A takes action aA,B takes actionaB

20. Preventing Profit Transfer (Cont’d) • Preventing the two approaches for convincing others: • Showing claimed cost. • One may update its claimed cost later. • Showing payment message. • Restricted verifier signature

21. Restricted Verifier Signature • The source node S signs its payment using our restricted verifier signature. • The restricted verifier signature can only be verified by the player node (payee) and a central bank. • The other player nodes have no way to verify the signature.

22. P,Q: large primes P=2Q+1 Z*p: multiplicative group mod P GQ: subgroup of quadratic residues ZQ: additive group mod Q g: enerator of GQ Each node vi has private key xiZQ,, and public key . A central bank has private key xBZQ,, and public key . Sign(): a standard digital signing algorithm Verify(): the corresponding verification algorithm H(): a cryptographic hash function Restricted Verifier Signature (Cont’d)

23. vi sign mthat can only be verified by vj and the bank. vj verify: Restricted Verifier Signature (Cont’d) Bank verify: r1,r2,r3 ZQ The signature is

24. Evaluations • Glomosim 2.03 • 100 nodes • Terrain 3000 X 3000 meters. • IEEE 802.11 (at 2Mbps) • Radio range 422.757 meters. Topology of the Randomly Generated Network.

25. Effect of Collusion (1) Utility of Each Colluding Node Minus Its Utility in the Nash Equilibrium.

26. Effect of Collusion (2) Distributions of the Number of Nodes Getting Less Utility in Collusion.

27. Computation Overhead Keysize: 1024 bits Hash function: SHA-1 Platform: Crypto++ 5.2.1 Laptop with 1.4GHz CPU and 768MB Memory

28. Conclusion • Admissible Group Strategyproofness cannot be achieved. • Admissible Strong Nash Equilibrium can be achieved. When our scheme is used, all Nash Equilibria are Admissible Strong Nash Equilibria. • Profit transfer can be prevented using our restricted verifier signature . • Simulation shows our scheme do prevent collusion and has good performance.

29. Thank you!