Randomized and Quantum Protocols in Distributed Computation Michael Ben-Or The Hebrew University Michael Rabin’s Birthday Celebration
Randomized Protocols Power of Randomization • Exponential speedup for known algorithms • Complexity – The jury is still out on this • Provably more powerful in distributed computation • Natural application: Use randomization for symmetry breaking Leader Election in anonymous networks Dining Philosophers Problem No deterministic solutions Choice Coordination, Mutual Exclusion Efficient randomized protocols Note: No deterministic solution No bounded-step, zero-error randomized solution
Quantum Protocols Quantum Leader Election in Anonymous Networks Tani, Kobayashi & Matsumoto A connected synchronous network of n identical processors. Processors can send/receive quantum messages to/from neighbors. Know just a bound N on number of processors, n ≤ N . This includes many cases with no deterministic solution. No quantum cheating - No initial entanglement not allowing special initial symmetric quantum state such as W 1 , 0 , , 0 0 , 1 , , 0 0 , 0 , , 1 n
Quantum Leader Election Cont. 1 • Each process prepares a register holding 0 1 2 • Compute quantumly if your value is the same as all other’s. n n n n n 2 2 0 1 1 x 0 x 1 • Measure “ Equal or Not” qubits. All the same value b . If b=0 measure value register. Only those processes with value=1 remain as potential leaders. If b=1 then global state is the symmetric state n 1 n 0 • Quantum Magic : Each process applies the same operation U n to its register and the resulting state is a supper position on some “ not all equal ” states.
Quantum Leader Election Cont. U k for even k is (assuming k contenders remain, initially k=n ) For odd k use V k (on two qubits) State transforms to a superposition on “not all equal” states. Keeping just max value guarantees less remaining contenders if k>1 . No need to know exact value k of remaining contenders. Letting k go down from N to 2 is good enough. After N-1 phases just one leader remains.
Randomized Protocols Cont. Byzantine Agreement: • n processes or players, P 1 ,… P n , each with an input bit b i • Want all non faulty players to reach agreement on a bit b such that All non faulty players agree on the same b If all P i start with the same b i then output b=b i We model faults by a computationally unbounded Adversary • Computer crash, no electricity – Fail-Stop fault model • Software or undetected hardware errors, incoherent or wrong data, malicious players – Byzantine fault model
BA Deterministic Protocols Assuming we have n players and at most t faults Protocols: • There are efficient deterministic t+1 rounds protocols tolerating t<n/3 Byzantine faults in the synchronous model [PSL77-78,GM93] Lower Bounds: • A deterministic lower bound of t+1 rounds for fail stop faults [FL82,DS81] • For Byzantine faults t<n/3 [PSL78]. • No deterministic protocols even for t=1 in the asynchronous setting [FLP82].
BA Randomized Protocols Weak Global Coin • We reduce agreement to weak global coin flipping • Decide when there is a large majority of players suggesting the same value b in {0,1} . n /2 n 0 ( n -3 t )/5 ( n - t )/2 ( n + t )/2 ( n +3 t )/5 Flip a coin Choose 1 Choose 0 prefer 0 prefer 1 • If the coin flip succeeds with probability p the expected number of round to reach agreement is O(1/ p ) .
BA Randomized Protocols Adversary can react to players’ random selections: • static or adaptive failures • private communication or full information about the system • fail stop or Byzantine type faults Examples: • Static, fail stop, full information adversary: Each player P i selects a random r i in [0, n 3 ). Declare the player with the min as the leader . Leader flips an unbiased coin. O (1) rounds protocol. • Adaptive, Byzantine, full information (even asynchronous) adversary: Use majority voting on random bits. Exp time, but just O (1) for t < O ( n 1/2 ).
BA Randomized Protocols – More Examples • Adaptive, fail-stop, full information adversary: Majority voting gives for all t < n n t / n log 2 t / n log( n ) matching the lower bound [BB98]. • Static, Byzantine, full information adversary: O(log n ) time protocols No lower bounds! • Coin Flipping with an Adaptive Adversary? All known robust coin flipping games select an almost random leader, and then the leader flips a coin. All this is useless in the adaptive setting. Are there better games than the “Majority” game for adaptive adversaries?
BA Randomized Protocols Rabin’s fast Byzantine agreement Why not hand out random “ global coins ” in advance , as part of the protocol’s description – O(1) expected time. To allow adaptive, Byzantine, private communication adversary , (knows the full state of the faulty players), use a digitally signed secret sharing scheme to store those global coins, revealing them only when needed. Verifiable Secret Sharing [CGMA] and Global Random Coin protocol from scratch (using BA). Replenish the stock of shared global coins in Rabin’s protocol when needed, using available previously prepared global coins.
BA Randomized Protocols Cont. Global Coins with Adaptive, Byzantine, private comm. adversary: Each player P i selects a random r i in [0,n 3 ). Declare the player with the min as the leader . Leader flips an unbiased coin. Problem: A bad player can choose 1 and get elected. First try: Independently for player P : Each P k , k=1 …n , selects random r i in [0,n 3 ), and set n r k (mod n 3 ) r = k=1 Problem: A bad player can select r k after other values are known and control r . Idea: Use Verifiable Secret Sharing (VSS) Problem: VSS requires Byzantine Agreement !? Idea :[FM88 ] A two round “weak agreement” protocol is good enough for here O(1) time protocol.
BA Quantum Protocols Byzantine Agreement in the Quantum World Adaptive, Byzantine, full information adversary: Players have pairwise quantum channels “ Full Information ” in the quantum setting: The adversary knows the description of the current pure state of the system. Toy Example: Adaptive, fail-stop, full information adversary Each player prepares 3 n 1 k , k , , k and a GHZ state k 0 0 , 0 , , 0 1 , 1 , , 1 and distributes the pieces to all n players.
BA Quantum Protocols 3 n 1 k , k , , k k 0 0 , 0 , , 0 1 , 1 , , 1 At the next round all players measure all the pieces they have; a leader is selected according to the shared minimum; and the corresponding measured bit serves as the “global coin”. Cor: We get an O(1) expected round agreement protocol. By delaying the measurements until all the quantum messages have arrived the adversary has to stop messages before the outcome is known, and so effectively the adaptive adversary isn’t stronger than the static one.
BA Quantum Protocols Adaptive, Byzantine, full information adversary: Idea: replace random shared secrets by a superposition on all possible n 3 secrets and all possible polynomials. • This is just an encoding of the superposition of all secrets using a standard CSS quantum error correcting code. • We can use the QVSS procedure of [CGS02] replacing Byzantine agreements with the “weak agreements” of [ FM88] • We get an O(1) round quantum Byzantine agreement protocol in the adaptive, Byzantine, full information adversary model, tolerating an optimal t<n/3 faults. • Works also in the recent Self Stabilizing Byzantine agreement protocols [BDH08, HBD10].
Quantum Protocols Byzantine Agreement @30 C&O@40 Open Problems • In the asynchronous setting we can handle only t<n/4 faults, while randomized BA is possible for t<n/3 . The classical “private channel” solution of [ CR93] uses secret authentication codes and this can’t work here. • Quantum Choice Coordination? Bounded register, zero-error, wait free… • Quantum Mutual Exclusion? Quantum version of [R80,RK92] protocol that tolerates a full information adversary? Quantum lower bound generalizing [KMRZ93]? • Quantum protocol verification tools.
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