Reaching Approximate Agreement in an Asynchronous Environment

1. Reaching Approximate Agreementin an Asynchronous Environment And what does it have to do with the Witness Protection Program

2. Today’s Lecture • What approximately happened at Byzantine • The Byzantine Agreement Problem • BA - Known Results • The Approximate Byzantine Agreement Problem • Formal Definition • Previous Work • Our Algorithm, Two Versions.

3. What “approximately” happened at Byzantine • May 29th, 1453 • The Turks are besieging the city of Constantinople, A.K.A Byzantine. • The Muslim generalsAre trying to coordinate an attack.

4. The Byzantine Agreement Problem • Introduced by Lamport, Pease and Shostak, 80-82 • A world of n processes/generals, t of them are faulty/traitors. The generals are trying to coordinate. • Can this be solved? • Depends on the model. • Computational Bounds / Cryptography • Network Topology • Synchronous VS. Asynchronous

5. BA - Known Results • Synchronous – 3t+1 deterministic algorithm • Asynchronous – No Deterministic Algorithm Exists – FLP 85 • Randomized Algorithms Exists – Benor, Bracha and more.

6. The Approximate Byzantine Agreement Problem • Introduced by Dolev et al, 82 • How “Approximate”? • Each process has a Real initial value • A predetermined Epsilon. • All processes must halt within Epsilon from each-other.

7. Formal Definition • Agreement - All non-faulty processes halt with values within Epsilon of each other; • Validity - The value of each non-faulty process must be within the range of the initial values of non-faulty processes. • Termination – All non-faulty processes must eventually halt.

8. Previous Work • Dolev. 82 • The family of algorithms. • Trimming Functions. • 3t+1 Synchronous • 5t+1 Asynchronous • Fekete – 86,94 • Work on asynchronous, failure-by-omission. • Proven Asymptotically optimal algorithms.

9. Our Algorithm • Using Reliable Broadcast with FIFO channels. • Correctness – If a non-faulty process p broadcasts m, all non-faulty processes will accept m from p. • Unforgeability – if a non-faulty process doesn’t broadcast m, no non-faulty process accept m from p • Relay – If a non-faulty process accepts m from p, all non-faulty processes eventually accept m from p. • Using Reliable Broadcast we can lower the requirement to 4t+1

10. Our Algorithm, n>4t

11. Our Algorithm, Cont… • The range of the non-faulty processes is cut in half in every round – intuition. • Note that each pair of processes have in common atleast 2t+1 values. • The worst the adversary can do, is “pick a side”. • After the trimming, there’s enough in common.

12. Our Algorithm, cont… • To reach 3t+1 we do the following: • Each process broadcasts its value • Collect values • Report what you’ve heard to all processes • Collects other’s reports. • When sufficient reports are obtained, • Trim the values, and calculate a new value

13. Our Algorithm, cont… What’s sufficient? • When we have n-t witnesses. • A witness for process q is a process p, whose first n-t values were also explicitly heard by q. • Common witnesses - Quorums

14. Our Algorithm, Cont… Initialization & Termination • Each round we trim the range in half. • Initial declarative round, where the bounds are set. • We run for log(range) rounds.

16. Our Algorithm, Cont… • The range of the non-faulty processes is cut in half in every round: • Every two processes have at least n-t common values. • The Median of the common values is remains after trimming in both processes. • Thus, after averaging, the range is cut.

17. Our Algorithm, Conclusion. • We have devised a t-resilient algorithm, where n>3t, and thus is Optimal • Convergence rate is bound by the non-faulty processes’ initial range. • The Witness concept may be useful for other problems as well.