The Secrecy of Compressed Sensing Measurements

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The Secrecy of Compressed Sensing Measurements

• Yaron Rachlin Dror Baron

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Compressed Sensing (CS) Secrecy Scenario

• Alice wants to send Bob secret message.
• Message is K-sparse.
• Alice uses CS projection matrix to encode
  message.
• Does matrix act as encryption key?
• If Bob knows CS matrix, can recover message.

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Compressed Sensing Attack Scenario

• Eve intercepts message, does not know matrix.
• Can Eve recover secret message?

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Is compressed sensing secure?

• Claims
• The encryption matrix can be viewed as a
  one-time pad that is completely secure I. Drori
  Compressed Video Sensing BMVA Symposium on 3D
  Video - Analysis, Display and Applications, 2008.
• effectively implements a weak form of encryption
  D. Baron, M. F. Duarte, S. Sarvotham, M. B.
  Wakin and R. G. Baraniuk An Information-Theoretic
  Approach to Distributed Compressed Sensing
  Allerton 2005.

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Notions of security

• Information theoretic H(messageciphertext)H(me
  ssage)
• Computationally unbounded adversary
• Computational
• Extracting message equivalent to solving
  computationally hard problem
• Computationally bounded adversary

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Perfect Secrecy?

• Definition of perfect secrecy (Shannon).
• X message, Y ciphertext, I(XY)0
• Does CS-based encryption achieve perfect secrecy?
  NO
• Noiseless case
• If message X0, ciphertext Y0.
• CS matrices satisfying RIP roughly preserve l2
  norm.
• Mutual information is positive.
• Could mutual information be small?

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Computational Secrecy

• Recovery is feasible, but hard for
  computationally bounded adversary. (Weaker)
• More widely used than perfect secrecy.
• How many matrices must an attacker try before
  finding the correct Phi matrix?
• Propose this as a computational notion of
  security for CS.

264 keys could be an unfortunate predicament.
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Application

• Example Biometrics
• Dont want to store lots of data in the clear.
• Can we just store features? (Reversible)
• If encryption key compromised, severe loss.
• Possible solution
• Compress (lossy, enable revocation)
• Then encrypt (high overhead)
• Or, compress encrypt in same step?
• Time critical application.

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Other Applications

• Low power sensors
• Sensor Networks nodes have limited battery life.
• Provides low-cost encryption while performing
  compression.
• High bandwidth sensors
• Networks of video cameras require low latency.

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Results

• Sender transmits
• Attacker guesses
• With probability one
• Theorem For randomly generated Gaussian ?, with
  MK1, each subset of M columns can be used to
  find an M-sparse x that will satisfy y ?x
  with probability one. For all subsets of size
  TltM, a T-sparse x will satisfy y ?x with
  probability zero.

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Strictly Sparse, Noiseless Case

• Intuition dim(subspace intersection) lt K.
• Pr(signal in intersection)0.
• M3, K2
• M3, K1

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Implications for secrecy

• Lemma With probability one, and will
  yield M-sparse solutions.
• What does result mean in terms of security?
• Information theoretic
• Can detect correct key
• Computational
• Need to evaluate (many) keys in ensemble until
  correct one found.

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Quality of Reconstruction

• True Signal. N376, K 37
• Attacker reconstruction using wrong matrix.
• Reconstruction with correct matrix.

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Simulations with L1 reconstruction

• Simulation of attacks using wrong measurement
  matrices.
• Best among 10,000 pairs gave significant error.
• Eve is in trouble!
• Bob reconstructs correctly.

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Other Settings

• Strictly Sparse, Noiseless (Results, Simulations)
• Compressible, Noiseless
• Strictly Sparse, Noisy (Ongoing Work)
• Compressible, Noisy
• Preliminary analysis indicates similar results
  feasible in other settings.

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• Thank you for your attention.
• Questions?