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.: AAA A AA A

2.  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. Compressed Sensing (CS) Secrecy Scenario 2

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

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. 4

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

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 l 2 norm.  Mutual information is positive.  Could mutual information be small? 6

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. 7 2 64 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. 8

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. 9

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. 10

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

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. 12

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

14. Simulations with L 1 reconstruction  Simulation of attacks using wrong measurement matrices.  Best among 10,000 pairs gave significant error.  Eve is in trouble!  Bob reconstructs correctly. 14

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. 15

16. Thank you for your attention. Questions? 16