Byzantine-Resilient Colorless - PowerPoint Presentation

Slide1

Byzantine-Resilient Colorless

Computaton

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Companion slides

forDistributed ComputingThrough Combinatorial TopologyMaurice Herlihy & Dmitry Kozlov & Sergio Rajsbaum

1

Slide2

Message-Passing

15-Feb-15

2

Distributed Computing through Combinatorial Topology

Slide3

Crash Failures

15-Feb-15

3

Distributed Computing through Combinatorial Topology

Slide4

Byzantine Failures

15-Feb-15

4

Distributed Computing through Combinatorial Topology

Slide5

He said, She said …

15-Feb-15

5

Distributed Computing through Combinatorial Topology

Slide6

He said, She said …

15-Feb-15

6

Distributed Computing through Combinatorial Topology

Slide7

Identifying Input Values

Distributed Computing through Combinatorial Topology7

sorry

t

Slide8

Identifying Input Values

Distributed Computing through Combinatorial Topology

8

!

t+1

Slide9

Nothing Learned

Distributed Computing through Combinatorial Topology9

t

t

t

…

each input reported by only

t

t

are silent

(dim

I

+ 1) t

No input learned if

n

+1

·

(dim

I

+2)

t

Slide10

means

t

is “pretty small”

Byzantine is harder!

requirement that:

RequirementDistributed Computing through Combinatorial Topology

10n+1 > (dim I + 2) t

dim

I

irrelevant for crash failures

Slide11

Strict Tasks

Distributed Computing through Combinatorial Topology11

¢

(

¾0) Å ¢(¾

1) = ¢(

¾0 Å ¾1)

usually require only

µ

Slide12

Main Result

Distributed Computing through Combinatorial Topology12

(

I

,O,¢) has a wait-free Byzantine protocol

…

if and only if …there is a continuous map

f

: |

skel

t

I

|



|O

|carried by

¢.where

n+1 > (dim I + 2) t

Slide13

Communication

Distributed Computing through Combinatorial Topology13

message-passing

asynchronous layers

Slide14

Messages

Distributed Computing through Combinatorial Topology14

(

P, tag, v)

sender

message type

sequence

of values

cannot be forged

can be forged

Slide15

Reliable Broadcast

Distributed Computing through Combinatorial Topology15

RBSend

(P, tag, v)

RBReceive(P, tag, v)

Slide16

Non-Faulty Integrity

Distributed Computing through Combinatorial Topology16

If non-faulty

P

never broadcasts (P,tag,v) …

Non-faulty

Q never receives (P,tag,v)via

RBSend

(

P, tag, v

)

via

RBReceive

(

P, tag, v

)

Slide17

Non-Faulty Liveness

Distributed Computing through Combinatorial Topology17

If non-faulty

P

does broadcast (P,tag,v) …

Every non-faulty

Q receives (P,tag,v)via

RBSend

(

P, tag, v

)

via

RBReceive

(

P, tag, v

)

Slide18

Global Uniqueness

Distributed Computing through Combinatorial Topology18

If non-faulty

Q,R

reliably receive…then tag

= tag’ and v

= v’even if P is faulty

(

P,tag,v

), (

P,tag’,v

’

)

respectively

…

Slide19

Global Liveness

Distributed Computing through Combinatorial Topology19

for non-faulty

Q,R

…then R

reliably receives (P,tag,v

)…even if P is faulty

if

Q

reliably receives

(

P,tag,v

)…

Slide20

Summary

Distributed Computing through Combinatorial Topology20

The only way ….

is by sending the

same fake value to everyone

a Byzantine process can misbehave ….

Slide21

RBSend Phase 1

Distributed Computing through Combinatorial Topology

21

(

P,

SEND,

v

)

Slide22

RBSend Phase 2

Distributed Computing through Combinatorial Topology

22

(

Q

,

P

,

ECHO,

v

)

first

(

P, SEND, v

)

?

background loop

Slide23

RBSend Phase 3

Distributed Computing through Combinatorial Topology

23

(

Q

,

P

,

READY,

v

)

1

st

time received

(

*,

P,

ECHO,v

)

from

n+1

-

t

?

background loop

Slide24

RBSend Phase 4

Distributed Computing through Combinatorial Topology

24

(

Q,

P

,

READY,

v

)

1

st

time received

(

*,

P,READY,v

)

from

t

+1

?

background loop

Slide25

RBSend Phase 5

Distributed Computing through Combinatorial Topology

25

deliver (

P,

v

)

1

st

time received

(

*,

P,READY,v

)

from

n-t

+1

?

Slide26

Lemma: Non-Faulty Integrity

Distributed Computing through Combinatorial Topology26

Suppose non-faulty

P

reliably receives(Q,v) from non-faulty Q

P

must have received n+1-t(*, READY, Q, v) messages

At least

n

+1-2

t

non-faulty

processes sent (

*,

ECHO,Q,v

)

Let R be the first

R must have received (Q,SEND,v) from Q

Slide27

Lemma: Non-Faulty Liveness

Distributed Computing through Combinatorial Topology27

Non-faulty

P

broadcasts (P,SEND,v)

eventually received by n+1-

tnon-faulty processeseach sends (*,ECHO,P,v)

eventually receives

n

+1-

t

(

*,

ECHO,P,v

)

each sends (

*,READY,P,v)

eventually receives n+1-

t (*,READY,P,v)

Slide28

Lemma: Global Uniqueness

Distributed Computing through Combinatorial Topology28

The

uniqueness tests

ensure that any processthat broadcasts (*,ECHO,P,v) or

(*,READY,P,v

)will not broadcast (*,ECHO,P,v') or (*,

READY,P,v

)

where

v



v’.

Slide29

Lemma: Global Liveness

Distributed Computing through Combinatorial Topology29

Non-faulty

Q

reliably receives (P, v) …

R another non-faulty process

Q received ¸ n+1-t (

*,

READY,P,v

)

if

¸

t

+1

non-faulty send (

*,READY,P,v)every non-faulty will receive & rebroadcast

every non-faulty eventually receives n+1-

t¸ n+1-2

t from non-faultyand delivers

Slide30

Reading a Value

Distributed Computing through Combinatorial Topology30

getQuorum

(Tag: tag):

Set of Message M := ;

while |M| <

n+1-t or trusted(M) =

;

do

upon

RBReceive

(

Q,tag,v

) do

M := M [ {(

Q,tag,v)} return M

Slide31

Reading a Value

Distributed Computing through Combinatorial Topology31

getQuorum

(Tag: tag):

Set of Message M := ;

while |

M| < n+1-t or trusted(M

) =

;

do

upon

RBReceive

(

Q,tag,v

) do M :=

M [ {(Q,tag,v

)} return M

returns set of messages (values)

Slide32

getQuorum

(Tag: tag): Set of Message

M :=

; while |M

| < n+1-

t or trusted(M) = ;

do

upon

RBReceive

(

Q,tag,v

) do

M := M

[ {(Q,tag,v)}

return MReading a ValueDistributed Computing through Combinatorial Topology32

wait to hear from all but t

Slide33

Reading a Value

Distributed Computing through Combinatorial Topology33

getQuorum

(Tag: tag): Set of Message

M := ;

while |M

| < n+1-t or trusted(M

)

=

;

do

upon

RBReceive

(

Q,tag,v) do M

:= M [ {(Q,tag,v

)} return M

values reported by t+1

Slide34

Reading a Value

Distributed Computing through Combinatorial Topology34

getQuorum

(Tag: tag): Set of Message

M := ;

while |M

| < n+1-t or trusted(M

) =

;

do

upon

RBReceive

(

Q,tag,v) do

M := M

[ {(Q,tag,v)}

return M

wait until you learn a trusted value

Slide35

Reading a Value

Distributed Computing through Combinatorial Topology35

getQuorum

(Tag: tag): Set of Message

M := ;

while |

M| < n+1-t or trusted(M

) =

;

do

upon

RBReceive

(

Q,tag,v) do

M := M

[ {(Q,tag,v)}

return M

safe to wait for both (why?)returns ¸

1 trusted values

Slide36

Set Agreement

Distributed Computing through Combinatorial Topology36

SetAgree

(v): value

RBSend(P,INPUT,v)

M: Set of Message :=

getQuorum(INPUT) return min Trusted(M)

Slide37

Set Agreement

Distributed Computing through Combinatorial Topology37

SetAgree

(v): value

RBSend(P,INPUT,v)

M: Set of Message :=

getQuorum(INPUT) return min Trusted(M)

at most

t

lower values missing

Slide38

Barycenrtric Agreement

Distributed Computing through Combinatorial Topology

38

Ri

:= (;,…, ;)M

i:= ;

Bi:= ;

R

i

[j]

= messages received by

P

i

from

P

j

M

i = messages received by Pi

Bi = Pi ‘s buddies

: processesthat reported same set of vertices

Slide39

Barycentric Agreement (1)

Distributed Computing through Combinatorial Topology

39

RBSend(

Pi,INPUT,v) M

i := getQuorum

(INPUT)reliably broadcast my input vertex

reliably receive others’ input vertices

Slide40

Barycentric Agreement (2)

Distributed Computing through Combinatorial Topology

40

background

// run forever upon RBReceive(P

j,INPUT,u)

do Mi := Mi

[

{u}

RBSend

(

Pi,REPORT,M

i

)

record newly-learned input vertices

forward set of vertices with REPORT tag

Slide41

Barycentric Agreement

Distributed Computing through Combinatorial Topology

41

while

|Bi| < n+1-t do

upon RBReceive

(Pj,REPORT,Mj) do Ri[j

] :=

M

j

B

i

:= {

P: Ri[l

] = Mi, 0 ·

l · n}return

trusted(Mi)

until I have n+1-t buddies…

accumulate new reports …

return set of trusted input vertices.

Slide42

Lemma

Distributed Computing through Combinatorial Topology42

Sequence of

M

i vertex sets reliably broadcast by Pi are monotonically increasing as sent & as received

When sent, by construction …

when received, because FIFO message channels

Slide43

Lemma: Protocol Terminates

Distributed Computing through Combinatorial Topology43

Suppose BWOC

P

i runs forever …Eventually

Mi assumes final value

M …Non-faulty Pj

,

with

M

j

,

receives

M

P

j must have sent

Mj to Pi

Mj ½ M

Mj = M

ORP

i will sent M to Pj

P

j has sent M to Pi

Slide44

All Mi

, Mj Totally Ordered

Distributed Computing through Combinatorial Topology

44

If Pi broadcasts M(0

), …, M(k

), then M(i) ½ M

(i+1

)

To decide …

P

i

received

M

i

from X, |X|

¸ n+1-t

Pj received Mj from

Y, |Y| ¸ n+1-t

some Pk 2 X Å Y sent both

M

i, Mjso

Mi, Mj are ordered.

Slide45

Lemma

Distributed Computing through Combinatorial Topology45

Non-faulty

P

i and Pj have|M

i Å

Mj| ¸ n+1-ttrusted(

M

i

Å

M

j

)

 ;Mi

µ Mj or Mj

µ Mi

proof in book

Slide46

Byzantine Computability

Distributed Computing through Combinatorial Topology46

(

I

,O,¢) has a

wait-free Byzantine protocol …

if and only if …there is a continuous map

f

: |

skel

t

I

|



|

O|carried

by ¢.if

n+1 > (dim I + 2) t

Slide47

Map Implies Protocol

Distributed Computing through Combinatorial Topology47

f

: |

skelt I|  |

O|

Á: BaryN skelt

I



O

Solve using …

barycentric

agreement

t

-set agreementcarried by

¢.

Slide48

Protocol implies Map

Distributed Computing through Combinatorial Topology

48

reduction to crash-failure model

Slide49

Lower Bound

Distributed Computing through Combinatorial Topology49

Á

:

I  O

carried

by ¢.A strict colorless task (I

,

O

,

¢

)

is

trivial

if ther

e is a simplicial map

(A trivial task can be solved without communication)

Slide50

Theorem

Distributed Computing through Combinatorial Topology50

If a strict colorless task

(

I,O,¢)has a protocol, where

n+1 ·

(dim I + 2) t then that task is trivial(A trivial task can be solved without communication)

Slide51

Proof

Distributed Computing through Combinatorial Topology51

Let

¾

= {v0, …, vd}

be a d-simplex of I

consider an execution where …Pi

has input

v

i

mod d+1

no failures

Slide52

Proof

Distributed Computing through Combinatorial Topology52

…

…

S = {

P

0

, …,

P

n

-t

}

T = {

P

n+1-t

, …,

P

n

}

no communication

P

i

decides

u

i

S

i

= {

P

|

P

in

S

has input

v

i

}

Slide53

Proof

Distributed Computing through Combinatorial Topology53

If

u

i 2 ¢(¾i

) and …

there is a unique minimal ¾i such that

u

i

2

¢

(

¾

i’) then …

ui 2 ¢

(¾i Å ¾i

’) so …ui

2 ¢(¾i)

Slide54

Proof

Distributed Computing through Combinatorial Topology54

If all

¾

i = {vi} then the task is trivial.

so some minimal

¾i = {vi ,vj

}

Every simplex

¿

such that

u

i

2

¢(

¿) contains vj

Slide55

Proof

Distributed Computing through Combinatorial Topology55

…

…

T

have input

v

i

no communication

P

i

still

decides

u

i

S

j

change inpu

t to

v

i

S

j

are Byzantine, still claim

v

j

u

i

2

¢

(

¾

i

n

{

v

j

})

contradiction

Slide56

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Distributed Computing through Combinatorial Topology

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Distributed Computing through Combinatorial Topology

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