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Introduction

Classical error correction

s

t

- Errors in some bits, locations unknown
- → Code across bits

Network error correction

[Yeung & Cai 06]

s2

t1

s1

t2

unknown erroneous links

network

- Errors in some links/packets, locations unknown
- → Code across links/packets

r2

- Given a network and error model
- What communication rates are feasible? (info theory)
- How to achieve with practical codes? (coding theory)

5

s2

1

t1

1

4

s1

t2

unknown erroneous links

network

r1

Background – network error correction

- Rich literature on single-source multicast with uniform errors
- All sinks demand the same information
- Equal capacity network links/packets, any z can be erroneous
- Various capacity-achieving codes, e.g. [Cai & Yeung 06, Jaggi et al. 08, Koetter & Kschischang 08] and results on code properties, e.g. [Yang & Yeung 07, Balli, Yan & Zhang 07, Prasad & Sundar Rajan 09]

- Generalizations
- Non-multicast demands, multiple sources, rateless codes, non-uniform link capacities
- New capacity bounding and coding techniques
- Applications
- Streaming (non-multicast nested network)
- Distribution of keys/decentralized data (multi-source network)
- Computationally limited networks (rateless codes)

- Non-multicast nested networks, streaming communication
- Multiple-source multicast, key distribution
- Rateless codes, computationally limited networks
- Non-uniform link capacities

- Non-multicast nested networks, streaming communication
- O. Tekin, S. Vyetrenko, T. Ho and H. Yao, "Erasure correction for nested receivers," Allerton 2011.
- O. Tekin, T. Ho, H. Yao and S. Jaggi, “On erasure correction coding for streaming,” ITA 2012.
- D. Leong and T. Ho, “Erasure coding for real-time streaming,” ISIT 2012.
- Multiple-source multicast, key distribution
- Rateless codes, computationally limited networks
- Non-uniform link capacities

www.mobileapptesting.com/testing-the-alien-hunting-android-app/2011/03/et-phone-home

Background - non-multicast

- Not all sinks demand the same information
- Capacity even without errors is an open problem
- May need to code across different sinks’ data (inter-session coding)
- Not known in general when intra-session coding suffices
- Non-multicast network error correction
- Capacity bounds from analyzing three-layer networks (Vyetrenko, Ho & Dikaliotis 10)
- We build on this work to analyze coding for streaming of stored and online content

Streaming stored content

Source

Forward error correction

packet erasure link

(unit size packets)

Packet erasures

x

x

m1

I1

m3

I3

m2

I2

Initial play-out delay

Decoding deadlines

Demanded information

Nested network model

Temporal coding problem

Spatial network problem

source

Unit

capacity

links

x

x

Unit size

packets

m1

I1

m2

I2

m3

I3

t1

t2

t3

Deadlines

Sinks

I1

I1, I2

I1, I2, I3

Demanded info

- Each sink sees a subset of the info received by the next (nested structure)
- Non-multicast demands

Nested network model

source

x

x

m1

I1

m2

I2

m3

I3

t1

t2

t3

I1

I1, I2

I1, I2, I3

Capacity outer bound

Capacity outer bound

Packet error/erasure correction

streaming code

Finite blocklength network error/

erasure correction code

Problem and results

Problem

- Given an erasure model and deadlines m1, m2, …, what rate vectors u1, u2, … are achievable?

Results

- We find the capacity and a simple optimal coding scheme for a uniform erasure model
- At most z erasures, locations unknown a priori
- We show this scheme achieves at least a guaranteed fraction of the capacity region for a sliding window erasure model
- Constraints on number of erasures in sliding windows of certain length
- Exact optimal scheme is sensitive to model parameters

Problem and results

Problem

- Given an erasure model and deadlines m1, m2, …, what rate vectors u1, u2, … are achievable?

Results

- We find the capacity and a simple optimal coding scheme for a uniform erasure model
- At most z erasures, locations unknown a priori
- We show this scheme achieves at least a guaranteed fraction of the capacity region for a sliding window erasure model
- Constraints on number of erasures in sliding windows of certain length
- Exact optimal scheme is sensitive to model parameters

z erasures – upper bounding capacity

t1

t2

t3

I1

I1, I2

I1, I2, I3

- Want to find the capacity region of achievable rates u1,u2,…,un
- We can write a cut-set bound for each sink:

u1 ≤ m1 ̶ z

u1+u2 ≤ m2 ̶ z

…

u1+u2+…+un ≤ mn ̶ z

- Can we combine bounds for multiple erasure patterns and sinks to obtain tighter bounds?

Cut-set combining procedure

- Example: m1=3,m2=5, m3=7, m4=11, z=1

12

1245

124567

u1+H(X1X2|I1) ≤2

u1+H(X1X3|I1) ≤2

u1+H(X2X3|I1) ≤2

13

1345

134567

- Obtain bounds involving progressively
- more links and rates ui, by iteratively applying steps:
- Extend: H(X |I1i-1)+ |Y| ≥ H(X,Y |I1i-1)= H(X,Y |I1i)+ ui
- where X,Y is a decoding set for Ii
- Combine:

1234567

23

2345

234567

1234

123467

1234567

1235

123567

123456

123457

123456

1234567

1234567

123457

1234567

Upper bound derivation graph

- Example: m1=3,m2=5, m3=7, m4=11, z=1

12

1245

124567

Capacity region:

u1≤2

3u1+2u2 ≤8

3u1+2u2+u3 ≤9

6u1+5u2+4u3 ≤24

6u1+4u2+2u3+u4≤20

9u1+6u2+4u3+3u4 ≤36

6u1+5u2+4u3+2u4 ≤28

6u1+4.5u2+4u3+3u4 ≤30

9u1+7.5u2+7u3+6u4 ≤54

13

1345

134567

1234567

23

2345

234567

1234

123467

1234567

1235

123567

123456

123457

123456

1234567

1234567

123457

1234567

Different choices of links at each step give different upper bounds

Exponentially large number of bounds

Only some are tight – how to find them?

We use an achievable scheme as a guide and show a matching upper bound

Intra-session Coding

- Separate erasure coding over each sink’s data
- Code design → capacity allocation problem
- yj,k : capacity onkthlink allocated to jth sink’s data
- We may assume yj,k = 0 for k>mj

- A rate vector (u1,u2,…,un) is achieved if and only if for every unerased set P:

“As uniform as possible” intra-session coding scheme

m1 = 10, m2 = 14, m3 = 18, m4 = 22,

u1 = 6, u2 = 3, u3 = 3, u4 = 4, z=2

0.75

0.25

0.25

0

0.5

0.5

0.1875

0.1875

0.1875

0.2

0

0

0.2

0.25

0.4

0.5

0.2

0.4

0.2

0.4

0.5

For a given rate vector, fill each row as uniformly as possible subject to constraints from previous rows

Example:

Can we do better?

Capacity region

Theorem: The z-erasure (or error) correction capacity region is achieved by the “as uniform as possible” coding scheme.

Characterization of the capacity region in a form that is simple to specify and calculate

Intra-session coding is also relatively simple to implement

Proof Idea

Consider any given rate vector (u1,u2,…,un) and let Ti,j denote its corresponding “as uniform as possible” allocation:

Show inductively: the conditional entropy of any set of unerased links given messages I1,…, Ik matches the residual capacity from the table

Use Ti,jvalues to find the appropriate path through upper bound derivation graph

Streaming online content

packet erasure link

(unit size packets)

Message creation

times (c=3)

Message decoding deadlines (d=8)

Messages arrive every c time steps at the source, and must be decoded within d time steps

Problem and results

Problem

- Given an erasure model and parameters c and d, what is the maximum size of independent uniform messages?

Results

- We find the capacity and a simple coding scheme that is asymptotically optimal for the following erasure models:
- #1: Limited number of erasures per sliding window
- #2: Erasure bursts and guard intervals of certain lengths
- For other values of burst length and guard interval, optimal inter-session convolutional code constructions [Martinian & Trott 07, Leong & Ho 12]

Problem and results

Problem

- Given an erasure model and parameters c and d, what is the maximum size of independent uniform messages?

Results

- We find the capacity and a simple coding scheme that is asymptotically optimal for the following erasure models:
- #1: Limited number of erasures per sliding window
- #2: Erasure bursts and guard intervals of certain lengths
- For other values of burst length and guard interval, optimal inter-session convolutional code constructions [Martinian & Trott 07, Leong & Ho 12]

Code construction

constant number of current messages at each time step

when d is a multiple of c …

messages 2, 3, 4 are current at t = 12

Divide each packet evenly among current messages

Intra-session coding within each message

Code construction

variable number of current messages at each time step

when d is not a multiple of c …

messages 3, 4, 5 are current at t = 13

messages 3, 4 are current at t = 12

Divide each packet evenly among current messages

Intra-session coding within each message

Capacity result

- Like the previous case, converse obtained by
- combining bounds for multiple erasure patterns and sinks (deadlines)
- inductively obtaining upper bounds on the entropy of sets of unerased packets, conditioned on previous messages
- The converse bound coincides with the rate achieved by our coding scheme asymptotically in the number of messages n
- Gap for small n corresponds to underutilization of capacity at the start and end by the time-invariant coding scheme

- Non-multicast nested networks, streaming communication
- Multiple-source multicast, key distribution
- T. Dikaliotis, T. Ho, S. Jaggi, S. Vyetrenko, H. Yao, M. Effros, J. Kliewer and E. Erez, "Multiple-access Network Information-flow and Correction Codes," IT Transactions 2011.
- H. Yao, T. Ho and C. Nita-Rotaru, "Key Agreement for Wireless Networks in the Presence of Active Adversaries,"Asilomar 2011.
- Rateless codes, computationally limited networks
- Non-uniform link capacities

- www.fanpop.com/spots/the-usual-suspects

Multiple-source multicast, uniform z errors

s2

s1

t

- Sources with independent information

We could partition network capacity among different sources…

But could rate be improved by coding across different sources? To what extent can different sources share network capacity?

Challenge: owing to the need for coding across sources in the network and independent encoding at sources, straightforward extensions of single-source codes are suboptimal

Related work: code construction in (Siavoshani, Fragouli & Diggavi 08) achieves capacity for C1+C2=C

Coherent (known topology) and noncoherent (unknown topology) cases

Multiple-source multicast, uniform z errors

s2

- Coherent (known topology) and noncoherent (unknown topology) cases

s1

t

Sources with independent information

We could partition network capacity among different sources…

But could rate be improved by coding across different sources? To what extent can different sources share network capacity?

Challenge: owing to the need for coding across sources in the network and independent encoding at sources, straightforward extensions of single-source codes are suboptimal

Related work: code construction in (Siavoshani, Fragouli & Diggavi 08) achieves capacity for C1+C2=C

Multiple-source multicast, uniform z errors

s2

- Coherent (known topology) and noncoherent (unknown topology) cases

s1

t

Sources with independent information

We could partition network capacity among different sources…

But could rate be improved by coding across different sources? To what extent can different sources share network capacity?

Challenge: owing to the need for coding across sources in the network and independent encoding at sources, straightforward extensions of single-source codes are suboptimal

Related work: code construction in (Jafari, Fragouli & Diggavi 08) achieves capacity for C1+C2=C

Capacity region

- Theorem: The coherent and non-coherent capacity region under any z link errors is given by the cut set bounds
- U= set of source nodes
- mS= min cut capacity between sources in subset S of Uand each sink
- ri= rate from the ithsource
- Redundant capacity can be fully shared via coding

Capacity-achieving non-coherent code constructions

- Probabilistic construction
- Joint decoding of sources, using injection distance metric
- Subspace distance metric used in single-source case is insufficient in multi-source case
- Lifted Gabidulin rank metric codes over nested fields
- Successive decoding of sources
- Linear transformation to separate out other sources’ interference increases the field size of errors
- Sources encode over nested extension fields

An application: key distribution

R

wants k1, k2, k3

V1

V2

V3

V4

V5

V6

V7

V8

V9

k1, k2

k1, k2

k1, k2

k1, k3

k1, k3

k1, k3

k2, k3

k2, k3

k2, k3

- Robust distribution of keys from a pool (or other decentralized data)
- Nodes hold subsets of keys, some pre-distributed
- Further exchange of keys among nodes
- Want to protect against some number of corrupted nodes
- Questions:
- How many redundant transmissions are needed?
- Can coding help?

An application: key distribution

R

wants k1, k2, k3

S1

S3

S2

V9

V1

V1

V2

V2

V3

V3

V4

V4

V5

V5

V6

V6

V7

V7

V8

V8

V9

R

k1, k2

k1, k2

k1, k2

k1, k3

k1, k3

k1, k3

k2, k3

k2, k3

k2, k3

Problem is equivalent to multi-source network error correction

Coding across keys strictly outperforms forwarding in general

- Non-multicast nested networks, streaming communication
- Multiple-source multicast, key distribution
- Rateless codes, computationally limited networks
- S. Vyetrenko, A. Khosla & T. Ho, “On combining information-theoretic and cryptographic approaches to network coding security against the pollution attack,” Asilomar 2009.
- W. Huang, T. Ho, H. Yao & S. Jaggi, “Rateless resilient network coding against Byzantine adversaries,” 2012.
- Non-uniform link capacities

Background – adversarial errors in multicast

- Information theoretic network error correction
- Prior codes designed for a given mincut and max no. of errors zU
- Achieve mincut-2zU, e.g. [Cai and Yeung 06, Jaggi et al. 08, Koetter & Kschischang 08]
- No computational assumptions on adversaries
- Use network diversity and redundant capacity as resources
- Cryptographic signatures with rateless network codes
- Signatures for checking network coded packets, e.g. [Charles et al. 06, Zhao et al. 07, Boneh et al. 09]
- Achieve realized value of mincut after erasures
- Use computation, key infrastructure as resources

Motivation

Cryptographic approach

- Does not require a priori estimates of network capacity and errors (rateless)
- Achieves higher rate
- Performing signature checks requires significant computation; checking all packets at all nodes can limit throughput if nodes are computationally weak, e.g. low-power wireless nodes

Questions:

- Can we achieve the rateless benefits without the computational drawback?
- Can we use both network diversity as well as computation as resources, to do better than with each separately?

Rateless network error correction codes

- Incrementally send redundancy until decoding succeeds
- Without an a priori bound on the number of errors, need a means to verify decoding
- We give code constructions using respectively:
- Shared secret randomness between source and sink (small compared to message size)
- Cryptographic signatures
- These constructions are asymptotically optimal:
- Decoding succeeds w.h.p. once received information/errors satisfy cut set bound
- Overhead becomes negligible with increasing packet length

Rateless code using shared secret

- Shared secret is random and independent of the message
- Non-rateless case [Nutman and Langberg 08]
- Redundancy Y added to message W so as to satisfy a matrix hash equation [Y W I ]V= H defined by shared secret (V, H)
- Hash is used to extract[Y W I ] from received subspace
- Challenges in the rateless case:
- Calculate redundancy incrementally such that it is cumulatively useful for decoding
- Send redundancy incrementally
- Growth in dimension of subspace to be recovered in turn necessitates more redundancy

Rateless code using shared secret

Message W

Short packets

Long packets

Linearly indep redundancy for additions

y1

1

d11y1

d11

C1 W

C1

y2

0 1

d21y1+

C2 W

C2

d21d22

Linearly dep redundancy for erasures

d22y2

y3

C3 W

C3

0 0 1

d31d32d33

d31y1+

C4 W

C4

d32y2+d33y3

Each adversarial error packet can correspond to an addition (of erroneous information) and/or an erasure (of good information)

Code structure:

yk = w V(k)+hk, wherew is the vectorized message, Vij(k) =akij, and hk and akare shared secrets

Rateless code using signatures

Message W

Y1

Linearly independent redundancy for additions

C1 W+D11 Y1

C1

D11

Linearly dependent redundancy for erasures

Y2

C2 W+D21 Y1+D22 Y2

C2

D21

D22

Each adversarial error packet can correspond to an addition (of erroneous information) and/or an erasure (of good information)

Code structure:

yi= w Si , wherew is the vectorized message and Siis a generic known matrix

Example: simple hybrid strategy on wireless butterfly network

Node D has limited computation and outgoing capacity

→ Probabilistically checks/codes a fraction of packets

- Proportion of packets checked/coded chosen to maximize expected information rate subject to computational budget

- Non-multicast nested networks, streaming communication
- Multiple-source multicast, key distribution
- Rateless codes, computationally limited networks
- Non-uniform link capacities
- S. Kim, T. Ho, M. Effros and S. Avestimehr, "Network error correction with unequal link capacities," IT Transactions 2011.
- T. Ho, S. Kim, Y. Yang, M. Effros and A. S. Avestimehr, "On network error correction with limited feedback capacity," ITA 2011.

http://www.geekosystem.com/tag/starbucks/

Uniform and non-uniform links

- Adversarial errors on any z fixed but unknown links
- Uniform links:
- Multicast error correction capacity = min cut – 2z
- Worst-caseerrors occur on the min cut
- Non-uniform links:
- Not obvious what are worst-case errors
- Cut size versus link capacities
- Feedback across cuts matters (can provide information about errors on upstream links)
- Related work: Adversarial nodes (Kosut, Tong & Tse 09)

Tighter cut set bounding approach

Zigzag network

- The classical cut set bound is equivalent to adding reliable, infinite-capacity bidirectional links between each pair of nodes on each side of the cut
- Tighter bounds can be obtained by taking into account which forward links affect or are affected by which feedback links
- Equivalent to adding a link (i,j) only if there is a directed path from node i to node j on that does not cross the cut

46

New cut-set bound

For any cut Q,

- adversary can erase a set of k ≤ z forward links
- adversary then chooses two sets Z1 ,Z2 of z-k links s.t. decoder cannot distinguish which set is adversarial:
- no feedback links downstream of Z1 ,Z2 ,
- downstream feedback links are included in Zi , or
- downstream feedback links Wi that are not in Zi have relatively small capacity s.t. distinct codewords have the same feedback link values
- sum of capacities of remaining forward links + capacities of links in W1 ,W2 is an upper bound

Bound is tight on some families of zigzag networks

Achievability - example

- For z=1, upper bound= 5
- Without feedback link, capacity = 2
- Can we use feedback link to achieve rate 5?

∞

z=1

Achieve rate 3 using new code construction

∞

“Detect and forward” coding strategy

a

+c

- Some network capacity is allocated to redundancy enabling partial error detection at intermediate nodes
- Nodes that detect errors forward additional information allowing the sink to locate errors
- Use feedback capacity to increase the number of symbols transmitted with error detection
- Remaining network capacity carries an MDS error correction code over all information symbols

b

a

∞

b

d

- e

c

r1

z=1

Achieve rate 3 using new code construction

∞

r2

z=1

capacity= 5

Conclusion

- Network error correction
- New coding and outer bounding techniques for non-multicast demands, multiple sources, non-uniform errors
- A model for analysis and code design in various applications, e.g. robust streaming, key distribution
- Rateless and hybrid codes for computationally limited networks with adversaries

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