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April 2008 1 Brahms Byzantine-Resilient Random Membership Sampling Bortnikov, Gurevich, Keidar, Kliot, and Shraer

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April 2008 2 Edward (Eddie) BortnikovMaxim (Max) GurevichIdit Keidar Gabriel (Gabi) Kliot Alexander (Alex) Shraer

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April 2008 3 Why Random Node Sampling Gossip partners Random choices make gossip protocols work Unstructured overlay networks E.g., among super-peers Random links provide robustness, expansion Gathering statistics Probe random nodes Choosing cache locations

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April 2008 4 The Setting Many nodes – n 10,000s, 100,000s, 1,000,000s, … Come and go Churn Every joining node knows some others Connectivity Full network Like the Internet Byzantine failures

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April 2008 5 Adversary Attacks Faulty nodes (portion f of ids) Attack other nodes May want to bias samples Isolate nodes, DoS nodes Promote themselves, bias statistics

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April 2008 6 Previous Work Benign gossip membership Small (logarithmic) views Robust to churn and benign failures Empirical study [Lpbcast,Scamp,Cyclon,PSS] Analytical study [Allavena et al.] Never proven uniform samples Spatial correlation among neighbors’ views [PSS] Byzantine-resilient gossip Full views [MMR,MS,Fireflies,Drum,BAR] Small views, some resilience [SPSS] We are not aware of any analytical work

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April 2008 7 Our Contributions 1. Gossip-based attack-tolerant membership Linear portion f of failures O(n 1/3 )-size partial views Correct nodes remain connected Mathematically analyzed, validated in simulations 2. Random sampling Novel memory-efficient approach Converges to proven independent uniform samples The view is not all bad Better than benign gossip

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April 2008 8 Brahms 1. Sampling - local component 2. Gossip - distributed component sample Sampler Gossip view

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April 2008 9 Sampler Building Block Input: data stream, one element at a time Bias: some values appear more than others Used with stream of gossiped ids Output: uniform random sample of unique elements seen thus far Independent of other Samplers One element at a time (converging) next sample Sampler

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April 2008 10 Sampler Implementation Memory: stores one element at a time Use random hash function h From min-wise independent family [Broder et al.] For each set X, and all, Sampler next sample init Keep id with smallest hash so far Choose random hash function

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April 2008 11 Component S: Sampling and Validation Sampler sample Sampler next init using pings id stream from gossip Validator S

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April 2008 12 Gossip Process Provides the stream of ids for S Needs to ensure connectivity Use a bag of tricks to overcome attacks

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April 2008 13 Gossip-Based Membership Primer Small (sub-linear) local view V V constantly changes - essential due to churn Typically, evolves in (unsynchronized) rounds Push: send my id to some node in V Reinforce underrepresented nodes Pull: retrieve view from some node in V Spread knowledge within the network [ Allavena et al. ‘05]: both are essential Low probability for partitions and star topologies

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April 2008 14 Brahms Gossip Rounds Each round: Send pushes, pulls to random nodes from V Wait to receive pulls, pushes Update S with all received ids (Sometimes) re-compute V Tricky! Beware of adversary attacks

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April 2008 15 Problem 1: Push Drowning Push Alice Push Bob Push Dana Push Carol Push Ed Push Mallory Push M&M Push Malfoy A D B E M M M M M

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April 2008 16 Trick 1: Rate-Limit Pushes Use limited messages to bound faulty pushes system-wide E.g., computational puzzles/virtual currency Faulty nodes can send portion p of them Views won’t be all bad

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April 2008 17 Problem 2: Quick Isolation Push Alice Push Bob Push Dana Push Carol Push Ed Push Mallory Push M&M Push Malfoy A C E M Ha! She’s out! Now let’s move on to the next guy! D

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April 2008 18 Trick 2: Detection & Recovery Do not re-compute V in rounds when too many pushes are received Slows down isolation; does not prevent it Push Mallory Push M&M Push Malfoy Hey! I’m swamped! I better ignore all of ‘em pushes… Push Bob

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April 2008 19 Problem 3: Pull Deterioration Pull M8M8 ABM1M1 M2M2 CDM3M3 M4M4 M7M7 EFM5M5 M6M6 E M3M3 M3M3 EM7M7 M8M8 50% faulty ids in views 75% faulty ids in views

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April 2008 20 Trick 3: Balance Pulls & Pushes Control contribution of push - α|V| ids versus contribution of pull - β|V| ids Parameters α, β Pull-only eventually all faulty ids Push-only quick isolation of attacked node Push ensures: system-wide not all bad ids Pull slows down (does not prevent) isolation

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April 2008 21 Trick 4: History Samples Attacker influences both push and pull Feedback γ|V| random ids from S Parameters α + β + γ = 1 Attacker loses control - samples are eventually perfectly uniform Yoo-hoo, is there any good process out there?

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April 2008 22 View and Sample Maintenance Pushed ids Pulled ids S |V| |V||V| |V||V| View VSample

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April 2008 23 Key Property Samples take time to help Assume attack starts when samples are empty With appropriate parameters E.g., Time to isolation > time to convergence Prove lower bound using tricks 1,2,3 (not using samples yet) Prove upper bound until some good sample persists forever Self-healing from partitions

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April 2008 24 History Samples: Rationale Judicious use essential Bootstrap, avoid slow convergence Deal with churn With a little bit of history samples (10%) we can cope with any adversary Amplification!

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April 2008 25 Analysis 1. Sampling - mathematical analysis 2. Connectivity - analysis and simulation 3. Full system simulation

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April 2008 26 Connectivity Sampling Theorem: If overlay remains connected indefinitely, samples are eventually uniform

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April 2008 27 Sampling Connectivity Ever After Perfect sample of a sampler with hash h: the id with the lowest h(id) system-wide If correct, sticks once the sampler sees it Correct perfect sample self-healing from partitions ever after We analyze PSP(t) – probability of perfect sample at time t

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April 2008 28 Convergence to 1 st Perfect Sample n = 1000 f = 0.2 40% unique ids in stream

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April 2008 29 Scalability Analysis says: For scalability, want small and constant convergence time independent of system size, e.g., when

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April 2008 30 Connectivity Analysis 1: Balanced Attacks Attack all nodes the same Maximizes faulty ids in views system-wide in any single round If repeated, system converges to fixed point ratio of faulty ids in views, which is < 1 if γ=0 (no history) and p < 1/3 or History samples are used, any p There are always good ids in views!

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April 2008 31 Fixed Point Analysis: Push i Local view node 1 Local view node i Time t: push 1 Time t+1: push from faulty node lost push x(t) – portion of faulty nodes in views at round t; portion of faulty pushes to correct nodes : p / ( p + ( 1 − p )( 1 − x(t) ) )

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April 2008 32 Fixed Point Analysis: Pull i Local view node 1 Local view node i Time t: pull from i: faulty with probability x(t) Time t+1: E[x(t+1)] = p / (p + (1 − p)(1 − x(t))) + ( x(t) + (1-x(t)) x(t) ) + γf pull from faulty

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April 2008 33 Faulty Ids in Fixed Point With a few history samples, any portion of bad nodes can be tolerated Perfectly validated fixed points and convergence Assumed perfect in analysis, real history in simulations

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April 2008 34 Convergence to Fixed Point n = 1000 p = 0.2 α=β=0.5 γ=0

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April 2008 35 Connectivity Analysis 2: Targeted Attack – Roadmap Step 1: analysis without history samples Isolation in logarithmic time … but not too fast, thanks to tricks 1,2,3 Step 2: analysis of history sample convergence Time-to-perfect-sample < Time-to-Isolation Step 3: putting it all together Empirical evaluation No isolation happens

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April 2008 36 Targeted Attack – Step 1 Q: How fast (lower bound) can an attacker isolate one node from the rest? Worst-case assumptions No use of history samples ( = 0) Unrealistically strong adversary Observes the exact number of correct pushes and complements it to α|V| Attacked node not represented initially Balanced attack on the rest of the system

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April 2008 37 Isolation w/out History Samples n = 1000 p = 0.2 α=β=0.5 γ=0 Depend on α,β,p Isolation time for |V|=60

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April 2008 38 Step 2: Sample Convergence Perfect sample in 2-3 rounds n = 1000 p = 0.2 α=β=0.5, γ=0 40% unique ids Empirically verified

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April 2008 39 Step 3: Putting It All Together No Isolation with History Samples Works well despite small PSP n = 1000 p = 0.2 α=β=0.45 γ=0.1

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April 2008 40 p = 0.2 α=β=0.45 γ=0.1 Sample Convergence (Balanced) Convergence twice as fast with

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April 2008 41 Summary O(n 1/3 )-size views Resist attacks / failures of linear portion Converge to proven uniform samples Precise analysis of impact of attacks

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April 2008 42 Balanced Attack Analysis (1) Assume (roughly) equal initial node degrees x(t) = portion of faulty ids in correct node views at time t Compute E[x(t+1)] as function of x(t), p, , , Result #1: Short-term Optimality Any non-balanced schedule imposes a smaller x(t) in a single round

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April 2008 43 Balanced Attack Analysis (2) Result #2: Existence of Fixed Point X^ E[x(t+1)] = x(t) = X^ Analyze X^ (function of p, , , ) Conditions for uniqueness For = =0.5, p < 1/3, exists X^ < 1 The view is not entirely poisoned – history samples are not essential Result #3: Convergence to fixed point From any initial portion < 1 of faulty ids From [ Hillam 1975 ] (sequence convergence)

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