Resilient Network Coding in the Presence of Eavesdropping Byzantine Adversaries
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Transcript of Resilient Network Coding in the Presence of Eavesdropping Byzantine Adversaries
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Resilient Network Coding in the Presence of Eavesdropping Byzantine Adversaries
Michael LangbergSidharth Jaggi
Open University of Israel
ISIT 2007
Tsinghua University
Hongyi Yao
Proof incorrect!
2009
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Network Codes Resilient to Eavesdropping andJamming
Michael Langberg
Sidharth Jaggi
ORHow to share a secretwhen a (bad) person’s listeningand trying to fool the receiver
Open University of IsraelTsinghua
University
Hongyi Yao
University of Campinas
DaniloSilva
NetCod 2010
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Multicast
ALL of Alice’sinformationdecodableEXACTLYbyEACH Bob
Network Model
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Multicast Network Model
ALL of Alice’sinformationdecodableEXACTLYbyEACH Bob
[ACLY00] With mixing, C = min{Ci} achievable[LCY02],[KM01],[JCJ03],[HKMKE03] Simple (linear) distributed codes suffice
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Problem!
Eavesdropped Links ZI
Attacked Links ZO
Corrupted links
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Existence proofs/Converses
Efficient random distributed schemes
Cryptographic schemes
Universal schemes
Privacy (Secrecy)
[CY02], …, [RS07], …
[FMSS04],… [OB08], … [SK08],…
Security (Error Correction)
[CY06]/[YC06], …, [M06], …
[JLKKHM07],… [CJL06], [GP06], [ZKMH07], …
[KK07], [SKK08]…
Background
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Setup
1. Scheme A B C2. Network
C3. Data A4. Code C5. Bad links C6. Randomness A7. Transmission A B C8. Decode B
Eureka
Eavesdropped links ZI
Attacked links ZO
Who knows what
Stage
Privacy
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Background Efficient algorithms [SK08] [JLKKHM07],
[SKK08] “Optimal” rates (Secrecy) C-ZI (Security) C-2ZO Poly-time Distributed End-to-end Packet-based Topology unknown a priori Information-theoretically secure, private
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Privacy at rate C-ZI ([CY02],…)
ZI random symbols
C-ZI rate
Network coding + one-time padOptimal
ZI eavesdropped links
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Security at rate C-2ZO (…,[SKK08])
Optimal rateR = C-2ZO
Error-correcting code
Network(Operator channel [KK07], Rank-metric codes [SKK08])
ZO corrupted links
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Privacy + Security at rate C-2ZO-ZI
Optimal rate for zero-error codes
Network error-correcting codeRate C-2ZO
ZI random symbols
Overall rate C-2ZO-ZI
[NY09], [SK10]
ε-error?
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Shared-secret Security at rate C-ZO
[JLKKHM07]
€
Y = TX + T 'Z = T T '[ ]XZ ⎡ ⎣ ⎢
⎤ ⎦ ⎥
€
XZ ⎡ ⎣ ⎢
⎤ ⎦ ⎥= T T '[ ]
−1Y
Invertible w.h.p. [HKMKE03]
C2 secret hashes of X
Linear list with C2 variables*
Secure transmission at rate C-ZO!
Secure + private transmission at rate C-ZO-ZI!
w.h.p., unique decoding!
* Different list-decoding used
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Upper bound: C-ZO-ZI
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Just one (secret) bit for Bob… [JL07]
??
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Just one (secret) bit for Bob… [YSJL10]
€
0 →R0 ⎡ ⎣ ⎢
⎤ ⎦ ⎥
€
1 →RR' ⎡ ⎣ ⎢
⎤ ⎦ ⎥
€
} ZI} C − ZI
€
} ZI} C − ZI €
Decode(Y) =0, rank(Y ) <C1, rank(Y ) =C ⎧ ⎨ ⎩
Privacy: [SK08] (Only sees ZI links)
Security:
€
R0 ⎡ ⎣ ⎢
⎤ ⎦ ⎥→
RR' ' ⎡ ⎣ ⎢
⎤ ⎦ ⎥, only ZO <C − ZI packets
€
w.h.p. RR' ⎡ ⎣ ⎢
⎤ ⎦ ⎥→
R' 'R' ' ' ⎡ ⎣ ⎢
⎤ ⎦ ⎥, doesn't know R, R'
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Questions?