Transcript Optimal Multicast Algorithms - Dept. of IE, CUHK Staff Web

Resilient Network Coding
in the presence of
Byzantine Adversaries
Sidharth Jaggi
Michelle Effros
Michael Langberg
Tracey Ho
Sachin Katti
Muriel Médard
Dina Katabi
Obligatory Example/History
s
[ACLY00]
b1
C=2
[ACLY00] Characterization
Non-constructive
b2
[LYC03], [KM02] Constructive (linear) Exp-time design
b1
b2
[JCJ03], [SET03] Poly-time design Centralized design
b1
b1 +b2
b2
b1
b1
b1+b
b
+b
2
1
2
t1
t2
(b1b,b
1 2)
(b1,b2)
[SET03] Gap provably exists
[HKMKE03], [JCJ03] Decentralized design
.
.
.
Tons of work
[This work] All the above, plus
security
E
V
E
R
B
E
T
T
E
R
Multicast Network Model
ALL of Alice’s
information
decodable
EXACTLY
by
EACH Bob
Simplifying assumptions
• All links unit capacity
•(1 packet/transmission) [GDPHE04],[LME04] – No intereference
• Acyclic network
Multicast Network Model
2
ALL of Alice’s
information
decodable
EXACTLY
by
EACH Bob
2
3
Upper bound for multicast capacity C,
C ≤ min{Ci}
[ACLY00] With mixing, C = min{Ci} achievable!
[LCY02],[KM01],[JCJ03],[HKMKE03] Simple (linear) distributed codes suffice!
Problem!
Corrupted
links
Eavesdropped
links
Attacked
links
Setup
Eurek
a
1.
2.
3.
4.
5.
6.
7.
8.
Stage
Scheme
Network
Message
Code
Bad links
Coin
Transmit
Decode
Privacy
Who
knows
what
ABC
C
A C
C
C
A
BC
B
Eavesdropped
links ZI
Attacked
links ZO
Results
First codes
 Optimal rates (C-2ZO,C-ZO)
 Poly-time
 Distributed
 Unknown topology
 End-to-end
 Rateless
 Information theoretically secure
 Information theoretically private
 Wired/wireless
[HLKMEK04],[JLHE05],[CY06],[CJL06],[GP06]
Error Correcting Codes
T
Y
X
Y=TX+E
Generator
matrix
(Reed-Solomon Code)
E
R=C-2ZO
Low-weight
vector
0
 
0
c 0 
 
0
1
 
Distributed multicast
[HKMKE03]

Alice: Sends packets.
I
Bob gets (Each column encoded with
same transform T)
TX

“Small” rate-loss
X
C packets

A
T
Now Bob knows T and can decode.
B2
What happens with errors?


What happens when we implement
previous distributed algorithm?
Key idea: think of Calvin's error as an
addition to original information flow.

Alice:
X
I
R packets

Calvin:
E1
E2
ZO packets

Bob:
TX +T’E1
T+T’E2
Bob:
•T,T’ are unknown.
•
E ,E are unknown.
C packets
•System is not linear.
•How can Bob recover X?
1
2
Overview
Step 1: Show how to construct system of
linear equations to help recover X.
Step 2: System may have many solutions.
Calvin
Need to add redundancy to X.

Alice:
X
I

Calvin:
E1
E2

Bob:
TXT +T’E
X+E1
T+T’E
T 2
E = T’(E1-E2X)
Step 1: “list decoding” will
work as long as R ≤ C-ZO.
Step 2: “unique decoding”
will need an additional
redundancy of ZO.
All in all: RB=2 C-2ZO.
B1
Properties of E

Alice:
X
I
R

Calvin:
E1
E2
ZO
•Col. in TX+E.
= col. of TX + col. of E.
•Claim 1: E has column rank
Z (=Calvin's strength).
•Proof: Follows from fact
O
TX +T’E1
Bob:
=

T+T’E2 C
T X+E
T
that Calvin controls ZO links.
•Claim 2: Columns of TX
and E span disjoint spaces.
•Proof: R≤C-Z , random
encoding.
+
=
O
Limited eavesdropping:
Calvin can only see the
information on ZI links
If ZI<C-ZO=R, can implement a
secret channel [JL07]
•
• Theorems



Scheme achieves rate C-2ZO (optimal)
 Step 1: list decode (R ≤ C-ZO)
 Step 2: unique decode (Redundancy = ZO)
Secret channel: Instead of Step 2, send hash of
X. Rate = C-ZO (optimal)
Limited Adversary: Calvin limited in
eavesdropping – can implement secret channel
and obtain rate C-ZO.
Summary
Rate
Thm 1 C-ZO
Conditions
Secret
Thm 2 C-2ZO Omniscient
Thm 3 C-ZO
Limited
Optimal rates
Poly-time
Distributed
Unknown topology
End-to-end
Rateless
Information theoretically secure/private
Wired/wireless