A New Approach to Lossy Compression and Applicationsto Security

Eva C. Song

Department of Electrical EngineeringPrinceton University

Joint work with: Paul Cuff and H. Vincent Poor

November 9, 2015

Overview

security

data compression

data transmission

1

23

45

6

7

1 compression/source coding2 transmission/channel coding3 security/cryptography4 rate-distortion based

information-theoreticsecrecy

5 joint source-channel coding6 traditional

information-theoretic secrecy7 joint source-channel

information-theoreticsecrecy

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Lossy compression

Low compression (high quality) JPEG High compression (low quality) JPEG

tradeoff between compression and qualitycommon in: audio, video, images, streaming, etcpopular technique: MP3, JPEG, MPEG-4, etcgood for data storage and transmission

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Looking through the engineering glass

Encoder DecoderX M Y

X : data sourceM: encoded message (used for storage or transmission)Y : reconstructed dataencoder/decoder: data encoding methods such as JPEG, MP3, MP4objective: (size(M), distance(X ,Y ))

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Information theory

Encoder fn Decoder gn

X n M Y n

Assumption 1 (general): known source distributionAssumption 2 (a bit less general and this work)

I i.i.d. source distributionI large n

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My contribution

Invented compressor: Likelihood EncoderAchieves best rate-distortion:

I point-to-point lossy compressionI multiuser lossy compressionI SECURITY

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Perfect secrecy

X n Encoder fn Decoder gn X̂ n

K ∈ [1 : 2nR0 ]

Eavesdropper

M ∈ [1 : 2nR ]

Theorem (Shannon)A rate pair (R,R0) is achievable under perfect secrecy if and only if

R ≥ H(X ),

R0 ≥ H(X ).

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What if we reduce key size?

not perfect secrecyhow “imperfect”?1nH(X n|M) < H(X )

I hard to interpretI what can the eavesdropper do with the information?

more practical metric for secrecy

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Rate-distortion based secrecy

X n Encoder fn Decoder gn Y n

K ∈ [1 : 2nR0 ]

PZn|M Zn

M ∈ [1 : 2nR ]

Average distortion for the legitimate receiver:

E[db(X n,Y n)] ≤ Db

Minimum average distortion for the eavesdropper:

minPZn|M

E [de(X n,Zn)] ≥ De

Conclusion: secrecy is almost FREE!

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Really FREE?

assumption: one attempt!one-bit secrecy

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Secure source coding with causal disclosure

X n Encoder fn Decoder gn Y n

K ∈ [2nR0 ]

Eavesdropper Zn

X t−1

t = 1, ..., nM ∈ [2nR ]

Average distortion for the legitimate receiver:

E [db(X n,Y n)] ≤ Db

Minimum average distortion for the eavesdropper:

min{PZt |MXt−1}nt=1

E [de(X n,Zn)] ≥ De

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About causal disclosure

Fully generalizes Shannon cipher systemCorresponding setting under noisy broadcast channels (physical layer)More about our work: http://www.princeton.edu/~csong

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