resilient network coding in the presence of byzantine adversaries
DESCRIPTION
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]. [ACLY00] Characterization Non-constructive. b 1. b 2. E V E R B - PowerPoint PPT PresentationTRANSCRIPT
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!
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]