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Resilient Network Coding in the presence of Byzantine Adversaries
Michelle Effros
Michael Langberg
Tracey Ho
Sachin Katti
Muriel Médard
Dina Katabi
Sidharth Jaggi
Obligatory Example/Historys
t1 t2
b1 b2
b2
b2
b1
b1 b1
b1 b1
b1 (b1,b2)
b1+b2
b1+b2b1+b2
(b1,b2)
[ACLY00] [ACLY00] Characterization Non-constructive
[LYC03], [KM02] Constructive (linear) Exp-time design
[JCJ03], [SET03] Poly-time design Centralized design
[HKMKE03], [JCJ03] Decentralized design
EVER
BETTER
.
.
.
C=2
[This work] All the above, plus security
Tons of work
[SET03] Gap provably exists
Multicast
Simplifying assumptions• All links unit capacity
•(1 packet/transmission)• Acyclic network
ALL of Alice’sinformationdecodableEXACTLYbyEACH Bob
Network Model
[GDPHE04],[LME04] – No intereference
Multicast Network Model
ALL of Alice’sinformationdecodableEXACTLYbyEACH Bob
3
2
2
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!
Eavesdropped links
Attacked links
Corrupted links
Setup
1. Scheme A B C2. Network
C3. Message A C4. Code C5. Bad links C6. Coin A7. Transmit B C8. Decode B
Eureka
Eavesdropped links ZI
Attacked links ZO
Who knows what
Stage
Privacy
ResultsFirst 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
Y=TX+E
Generator matrix
Low-weightvector
YX
(Reed-Solomon Code)
1
0
0
0
0
c
T
E R=C-2ZO
Alice: Sends packets.
Bob gets (Each column encoded with same transform T)
Now Bob knows T and can decode.
Distributed multicastA
B2
X I
TX T
C packets
“Small” rate-loss
[HKMKE03]
What happens when we implement previous distributed algorithm?
Key idea: think of Calvin's error as an addition to original information flow.
Alice:
Calvin:
Bob:C packets
ZO packets
What happens with errors?
X I
TX T+T’E1 +T’E2
E1 E2
Bob:
•T,T’ are unknown.
•E1,E2 are unknown.
•System is not linear.
•How can Bob recover
X?
R packets
Alice:
Calvin:
Bob:
Overview
B1B2
X I
TX T
Calvin
+T’E1 +T’E2
E1 E2
Step 1: Show how to construct system of
linear equations to help recover X.
Step 2: System may have many solutions.
Need to add redundancy to X.
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: R = C-2ZO.
X+
= T’(E1-E2X)
Alice:
Calvin:
Bob:
+T’E2+T’E1
Properties of X I
E1 E2
X+
•Col. in X+.
= col. of X + col. of .
•Claim 1: has column rank ZO (=Calvin's strength).
•Proof: Follows from fact that Calvin controls ZO links.
•Claim 2: Columns of X and span disjoint spaces.
•Proof:R≤C-ZO, random encoding.
TTX
=+ =
R
ZO
C
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.
Limited eavesdropping:
•Calvin can only see the information on ZI links
•If ZI<C-ZO=R, can implement a secret channel [JL07]
SummaryRate Conditions
Thm 1 C-ZO Secret
Thm 2 C-2ZO Omniscient
Thm 3 C-ZO Limited
Optimal rates Poly-timeDistributedUnknown topologyEnd-to-endRatelessInformation theoretically secure/privateWired/wireless