The Secrecy of Compressed Sensing Measurements

1. The Secrecy of Compressed Sensing Measurements Yaron Rachlin & Dror Baron TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAA

2. 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.

3. Compressed Sensing Attack Scenario • Eve intercepts message, does not know matrix. • Can Eve recover secret message?

4. 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.

5. Notions of security 5 • Information theoretic – H(message|ciphertext)=H(message) • Computationally unbounded adversary • Computational – Extracting message equivalent to solving computationally hard problem • Computationally bounded adversary

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

7. 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.

8. Application • Example: Biometrics • Don’t 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.

9. 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.

10. Results • Sender transmits: • Attacker guesses: • With probability one: • Theorem: For randomly generated Gaussian ’, with M≥K+1, 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 T<M, a T-sparse x’ will satisfy y = ’x’ with probability zero.

11. Strictly Sparse, Noiseless Case • Intuition – dim(subspace intersection) < K. • Pr(signal in intersection)=0. • M=3, K=2 • M=3, K=1

12. 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.

13. Quality of Reconstruction • True Signal. N=376, K = 37 • Attacker reconstruction using wrong matrix. • Reconstruction with correct matrix.

14. 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.

15. 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.

16. Thank you for your attention. Questions?