1. QUORUMS By gil ben-zvi

2. definition Assume a universe U of servers, sized n. A quorum system S is a set of subsets of U, every pair of which intersect, each Q belongs to S is called a quorum.

3. EXAMPLES Weighted majorities: assume that every server s in the universe U is assigned a number of votes w(s). Then weighted majorities is a quorum set defined by

4. EXAMPLES MAJORITIES : a weighted majorities quorum system when all weights are the same. Singleton: a weighted majorities quorum system when for one server s: w(s)=1, and for each v of the other servers w(v)=0. (only quorum is s)

5. EXAMPLES Grid: suppose n is a square of some integer k. arrange the universe in a k x k grid. A quorum is the union of a full row and one element from each row below. FPP: suppose a projective plane over a field sized q. each point is an element, and each line is a quorum. By projective plane attributes, each quorum intersect.

6. More definitions Coterie: a coterie S is a quorum system such that for any Q1,Q2 quorums in S: Q1 isn’t included in Q2 Domination: coterie S1 dominates coterie S2 if for every quorum Q2 belongs to S2, there exist Q1 in S1, such that S1 is contained in S2. Strategy: a probability vector representing the probability to access each quorum.

7. measures Load: the load L(S) of a quorum system is the minimal access probability minimized over the strategies. Resilience: resilience is k, if k is the largest number such that for every k server crashes, one quorum remains unhit.

8. measures Failure probability: if every server has certain probability to crash (assuming independently here), the probability that each quorum is hit. Usually assuming each server has the same crash probability p.

9. Measures examples Singleton: load=1, resilience=0, failure probability=p Majorities: load is about ½. Resilience about (n-1)/2. failure probability (if p < ½) smaller than exp(e,-n). Grid: load is O(1/sqrt(n)). Resilience = sqrt(n)-1, failure probability tends to 1 as n grows.

10. Access protocol Implements the semantics of a multi-writer multi-reader atomic variable. Assumes all clients and servers are non byzantine, unique timestamp for a client Write: a client asks some quorum to obtain a set of value/timestamps pairs, then he writes his value with higher timestamp than each of the timestamps received to each server in the quorum.

11. Access protocol Read: a client asks for each server in some quorum to obtain a set of value/timestamp. The client chooses the pair with the highest timestamp. It writes back the pair to each server in some quorum Server S updates a pair of value/timestamp, only if the timestamp is greater than the timestamp currently in S

12. Byzantine quorum systems We will use access protocol to demonstrate the subject Assuming communication is reliable, clients are correct, servers can be byzantine, assuming that a non-empty set of subsets of U: BAD, is known, some B in BAD contains all the faulty servers.

13. Masking quorum systems A quorum system S is a masking quorum system for a fail-prone system BAD if the following properties are satisfied:

14. Access protocol write: remains the same Read: for a client to read the variable x, it queries servers for some quorum Q to obtain a set of value/timestamp pairs

15. Access protocol The client chooses the pair with the highest timestamp in C, or null if C is empty.

16. Access protocol Claim: a read operation that is concurrent with no write operations return the value written by the last preceding write operation in some serialization of all preceding write operations. Claim: there exists a masking quorum system for BAD iff is a masking quorum system for BAD

17. Access protocol Criterion: there exists a masking quorum system for BAD iff for all

18. F-masking quorum systems F-masking quorum system: A masking quorum system where BAD is the set of all groups of servers sized f. By previous claims: There exists a masking quorum system for BAD iff n>4f Each pair of quorums must intersect by at least 2f+1 elements.

19. examples For f-masking quorums:

20. Dissemination quorum systems Assumes clients can digitally sign the value/timestamp they propagate. Therefore weaker demands than masking A quorum system S is a dissemination quorum system for a fail-prone system BAD if the following properties are satisfied:

21. Dissemination quorum systems The same way as masking we reach the (different) criterion: There exists a dissemination quorum system for BAD iff If no more than f servers can fail, but any set of f servers can fail, then must hold: n>3f

22. Opaque masking quorum systems Motivation: We want not to expose the fail- prone system BAD. done by majority decision. properties for quorum system to become opaque masking system:

23. Opaque masking quorum systems Read: the modification is that the client choose the pair that appears most often, if there are multiple such sets, it chooses the newest one. Claim: Suppose maximum f servers can fail, there exists an opaque quorum system for BAD iff n>=5f, sufficient because quorums sized [(2n+2f)/3] is an opaque quorum system for B.

24. Opaque masking quorum system Claim: The load of any opaque system is at least ½. Proof: if we sum up the load of a certain quorum, we’ll get it’s bigger than it’s size/2. the claim follows. Example: hadamard matrix, world of size exp(2,l)

25. Faulty clients Solves the problem that a client will try to fail the protocol. The treatment here provides a single- writer multi-reader semantics. The write operation starts when the 1 st server receives update request, and ends when the last server sent acknowledgment.

26. Faulty clients Write: for a client c to write the value v, it chooses legal timestamp, larger than any timestamp it has chosen before, chooses a quorum Q, And then it sends to each server in Q, if after some timeout period it has not received acknowledgment, than it chooses another quorum.

27. Faulty clients-servers protocol The servers protocol is as follows: 1.if a server receives from a client c, with legal timestamp, then it sends to each member of Q. 2.If a server receives identical echo messages from every server in Q, then it sends to each member of Q.

28. Faulty clients-servers protocol 3. If a server receives identical ready messages from a set of servers that certainly doesn’t contain faulty server, it sends to Q. 4. If a server receives identical ready messages from a set Q1 of servers, such that Q1=Q\B for some B in BAD, it sends acknowledgment for c, and update the pair if t is greater than the timestamp it currently has.

29. Faulty servers-properties Agreement: if a correct server delivers and a correct server delivers then r=v Proof: if a correct server delivers, then echo must have been send by all correct servers in Q1. same about Q2, they intersect in a correct server, which doesn’t send different value with the same timestamp

30. Faulty servers-properties Claim: Read received last written value if it’s not concurrent with write operations. Proof: same as masking quorum system. Propagation: similar ideas to r.b, and byzantine agreement, if server decides to deliver, it is promised that all other decides that too. Validity: at the end a correct quorum will be accessed, so the write can end.

31. Load, capacity, availability Load: we will mark L(S), definition as before availability: failure probability with the same “p” for all the servers, we will mark it as Fp(S) Capacity: we’ll define a(S,k) as the maximum number of quorum accesses that S can handle during a period of k time units. Capp(S) is the limit of a(S,k)/k as k tends to infinity.

32. Load, capacity, availability Example: majorities The claim is that cap(S)=1/L(S), and there is a trade off between good availability and good load.

33. definitions The cardinality of the smallest quorum is denoted by c(S) The degree of an element i in a quorum system S is the number of quorums that contain i Let S be a quorum system. S is a s- uniform if |Q| = s for each Q in S S is (s,d) fair if it is s-uniform and deg(i)=d foreach i, it is called s-fair if it is (s,d) fair for some d.

34. LP We can use a linear programming to calculate the load and the strategy achieving the load.

35. DUAL LP Some time we want to use the dual linear program, in which we give probabilities over the elements of the world. It is a known fact that DLP<=LP

36. The load with failures A configuration is a vector in which it holds 1 in places representing the failing elements in the world Dead(x) is the group of elements failed, live(x) is the non failed ones S(x) is the sub collection of functioning quorums

37. Load with failiures The load of quorum system S over a configuration x, if S(x) is empty then L(S(x)) = 1, if there are functioning quorums we define it in similar way as before by linear programming problem. Let the elements fail with probabilities P=(p1,………,pn). Then the load is a random variable Lp(S) defined by:

38. Load with fails Claim: E(Lp(S))>=Fp(S) Claim: If (configurations) x>=z than L(S(x))>=L(S(z)) Proof: S(z) contains S(x), strategy for S(x) is for S(z) too. Claim: E(Lp(S)) is a non decreasing function.

39. Properties of the load Claim: L(S)>=c(S)/n Claim: L(S)>=1/c(S) Proof: if we choose probability 1/c(S) for every element in c(S) and 0 in the rest, we achieve possible solution for the DLP problem. Conclusion: L(S)>1/sqrt(n) (achieved when c(S) is close to sqrt(n)

40. Load/fail probability trade off Claim: Fp(S)>=exp(p,n*L(S)) Proof: the probability that all the elements in the smallest quorum will fail, (and therefore the quorum system fails) = exp(p,c(S)). Since c(S)<=nL(S) the claim follows.

41. examples Optimal load, optimal load/ failure tradeoff, good failure load – paths system B-grid system SC-grid system AndOr system

42. Load analyses Claim: Non dominated coteries have lower bounds. The claim follows if you choose strategy for the dominator by giving the probability only in quorums which contained by a quorum in the dominated quorum system Claim: voting systems have high load (more than ½)

43. Last slide!!!! Proof: if we define V=the sum of all votes (Vi), then the vector Yi=Vi/V is a solution for DLP larger than ½.