Randomized and Quantum Protocols in Distributed Computation

Transcription

1 Radomized ad Quatum Protocols i Distributed Computatio Michael Be-Or The Hebrew Uiversity Michael Rabi s Birthday Celebratio

2 Radomized Protocols Power of Radomizatio Expoetial speedup for kow algorithms Complexity The jury is still out o this Provably more powerful i distributed computatio Natural applicatio: Use radomizatio for symmetry breakig Leader Electio i aoymous etworks Diig Philosophers Problem No determiistic solutios Choice Coordiatio, Mutual Exclusio Efficiet radomized protocols Note: No determiistic solutio No bouded-step, zero-error radomized solutio

3 Quatum Protocols Quatum Leader Electio i Aoymous Networks Tai, Kobayashi & Matsumoto A coected sychroous etwork of idetical processors. Processors ca sed/receive quatum messages to/from eighbors. Kow just a boud N o umber of processors, N. This icludes may cases with o determiistic solutio. No quatum cheatig - No iitial etaglemet ot allowig special iitial symmetric quatum state such as W 1,0,,0 0,1,,0 0,0,,1

4 Quatum Leader Electio Cot. Each process prepares a register holdig Compute quatumly if your value is the same as all other s. 0 1 Measure Equal or Not qubits. All the same value b. If b=0 measure value register. Oly those processes with value=1 remai as potetial leaders. If b=1 the global state is the symmetric state x1 x Quatum Magic: Each process applies the same operatio U to its register ad the resultig state is a supper positio o some ot all equal states.

5 Quatum Leader Electio Cot. U k for eve k is (assumig k coteders remai, iitially k=) For odd k use V k (o two qubits) State trasforms to a superpositio o ot all equal states. Keepig just max value guaratees less remaiig coteders if k>1. No eed to kow exact value k of remaiig coteders. Lettig k go dow from N to 2 is good eough. After N-1 phases just oe leader remais.

6 Radomized Protocols Cot. Byzatie Agreemet: processes or players, P 1, P, each with a iput bit b i Wat all o faulty players to reach agreemet o a bit b such that All o faulty players agree o the same b If all P i start with the same b i the output b=b i We model faults by a computatioally ubouded Adversary Computer crash, o electricity Fail-Stop fault model Software or udetected hardware errors, icoheret or wrog data, malicious players Byzatie fault model

7 BA Determiistic Protocols Assumig we have players ad at most t faults Protocols: There are efficiet determiistic t+1 rouds protocols toleratig t</3 Byzatie faults i the sychroous model [PSL77-78,GM93] Lower Bouds: A determiistic lower boud of t+1 rouds for fail stop faults [FL82,DS81] For Byzatie faults t</3 [PSL78]. No determiistic protocols eve for t=1 i the asychroous settig [FLP82].

8 BA Radomized Protocols Weak Global Coi We reduce agreemet to weak global coi flippig Decide whe there is a large majority of players suggestig the same value b i {0,1}. /2 0 (-3t)/5 (-t)/2 (+t)/2 (+3t)/5 Choose 0 prefer 0 Flip a coi prefer 1 Choose 1 If the coi flip succeeds with probability p the expected umber of roud to reach agreemet is O(1/p).

9 BA Radomized Protocols Adversary ca react to players radom selectios: static or adaptive failures private commuicatio or full iformatio about the system fail stop or Byzatie type faults Examples: Static, fail stop, full iformatio adversary: Each player P i selects a radom r i i [0, 3 ). Declare the player with the mi as the leader. Leader flips a ubiased coi. O(1) rouds protocol. Adaptive, Byzatie, full iformatio (eve asychroous) adversary: Use majority votig o radom bits. Exp time, but just O(1) for t < O( 1/2 ).

10 BA Radomized Protocols More Examples Adaptive, fail-stop, full iformatio adversary: Majority votig gives for all t < t / log matchig the lower boud [BB98]. Static, Byzatie, full iformatio adversary: O(log ) time protocols No lower bouds! 2 t / log( ) Coi Flippig with a Adaptive Adversary? All kow robust coi flippig games select a almost radom leader, ad the the leader flips a coi. All this is useless i the adaptive settig. Are there better games tha the Majority game for adaptive adversaries?

11 BA Radomized Protocols Rabi s fast Byzatie agreemet Why ot had out radom global cois i advace, as part of the protocol s descriptio O(1) expected time. To allow adaptive, Byzatie, private commuicatio adversary, (kows the full state of the faulty players), use a digitally siged secret sharig scheme to store those global cois, revealig them oly whe eeded. Verifiable Secret Sharig [CGMA] ad Global Radom Coi protocol from scratch (usig BA). Repleish the stock of shared global cois i Rabi s protocol whe eeded, usig available previously prepared global cois.

12 BA Radomized Protocols Cot. Global Cois with Adaptive, Byzatie, private comm. adversary: Each player P i selects a radom r i i [0, 3 ). Declare the player with the mi as the leader. Leader flips a ubiased coi. Problem: A bad player ca choose 1 ad get elected. First try: Idepedetly for player P: Each P k, k=1, selects radom r i i [0, 3 ), ad set r = k=1 r k (mod 3 ) Problem: A bad player ca select r k after other values are kow ad cotrol r. Idea: Use Verifiable Secret Sharig (VSS) Problem: VSS requires Byzatie Agreemet!? Idea:[FM88] A two roud weak agreemet protocol is good eough for here O(1) time protocol.

13 BA Quatum Protocols Byzatie Agreemet i the Quatum World Adaptive, Byzatie, full iformatio adversary: Players have pairwise quatum chaels Full Iformatio i the quatum settig: The adversary kows the descriptio of the curret pure state of the system. Toy Example: Adaptive, fail-stop, full iformatio adversary Each player prepares ad a GHZ state 1 k0 ad distributes the pieces to all players. 3 k, k,, k 0,0,,0 1,1,,1

14 BA Quatum Protocols 3 1 k0 k, k,, k 0,0,,0 1,1,,1 At the ext roud all players measure all the pieces they have; a leader is selected accordig to the shared miimum; ad the correspodig measured bit serves as the global coi. Cor: We get a O(1) expected roud agreemet protocol. By delayig the measuremets util all the quatum messages have arrived the adversary has to stop messages before the outcome is kow, ad so effectively the adaptive adversary is t stroger tha the static oe.

15 BA Quatum Protocols Adaptive, Byzatie, full iformatio adversary: Idea: replace radom shared secrets by a superpositio o all possible 3 secrets ad all possible polyomials. This is just a ecodig of the superpositio of all secrets usig a stadard CSS quatum error correctig code. We ca use the QVSS procedure of [CGS02] replacig Byzatie agreemets with the weak agreemets of [FM88] We get a O(1) roud quatum Byzatie agreemet protocol i the adaptive, Byzatie, full iformatio adversary model, toleratig a optimal t</3 faults. Works also i the recet Self Stabilizig Byzatie agreemet protocols [BDH08, HBD10].

16 Quatum Protocols Byzatie Ope Problems I the asychroous settig we ca hadle oly t</4 faults, while radomized BA is possible for t</3. The classical private chael solutio of [CR93] uses secret autheticatio codes ad this ca t work here. Quatum Choice Coordiatio? Bouded register, zero-error, wait free Quatum Mutual Exclusio? Quatum versio of [R80,RK92] protocol that tolerates a full iformatio adversary? Quatum lower boud geeralizig [KMRZ93]? Quatum protocol verificatio tools.