Performance-based Security for Encoding of Information Signals. FA ( ) Paul Cuff (Princeton University)

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1 Performance-based Security for Encoding of Information Signals FA ( ) Paul Cuff (Princeton University)

2 Contributors Two students finished PhD Tiance Wang (Goldman Sachs) Eva Song (Amazon) Three current PhD students Sanket Satpathy Jingbo Liu Lanqing Yu Collaborators Vince Poor (Princeton) Sergio Verdu (Princeton) Haim Permuter (Ben Gurion) Ziv Goldfeld (Ben Gurion)

3 Encoding Signals Signal (sensor) Signal (control) Digital Encoding Adversary Attack Signal

4 Information Theoretic Security Began with Shannon s 49 paper Fewer assumptions than cryptography (omnipotent adversary) More resources required for mathematical guarantees Enabled by quantum physics Gives deep understanding of information

5 Within Information Theory Channel Capacity Source Coding Secure Channel Coding Secure Source Coding

6 Publications Journal Papers: Key Capacity for Product Sources with Application to Stationary Gaussian Processes (IEEE Trans. Inf. Theory) The Likelihood Encoder for Lossy Compression (IEEE Trans. Inf. Theory) The Henchman Problem: Measuring Secrecy by the Minimum Distortion in a List (to appear, IEEE Trans. Inf. Theory) Semantic-Security Capacity for Wiretap Channels of Type II (to appear, IEEE Trans. Inf. Theory) Conference Papers: Semantic-Security Capacity for Wiretap Channels of Type II Soft Covering with High Probability Key Generation with Limited Interaction Brascamp-Lieb Inequalities: An Information Theoretic View Smoothing Brascamp-Lieb Inequalities and Strong Converses for Common Randomness Generation Semantic-Security Capacity for the Physical Layer via Information Theory Strong Secrecy for Cooperative Broadcast Channels A Stronger Soft-Covering Lemma and Applications

7 Semantic Security in Wiretap Channels

8 The Setting: 1975 Wyner publishes five paper I will discuss two Wiretap Channel Common Information

9 Wiretap Channel Foundation of physical-layer security

10 Wiretap Channel M (nr bits) Enc X n Memoryless P Y,Z X Y n Dec Z n Secrecy Capacity: - Reliable communication - Z n contains no information about M

11 Solutions Wyner gave solution for degraded channels Csiszár-Körner gave solution for all channels (1978) Encoding requires pre-channel

12 Solution Degraded: C s = max Px I(X;Y) - I(X;Z) General: C s = max Pxu I(U;Y) - I(U;Z) U P X U X P Y,Z X Y Z

13 Encoding Same construction as point-to-point Codebook generated according to P X Send two messages, M and M M is random garbage The rate of M is I(X;Z)

14 Encoding Diagram M (nr bits) Enc X n Memoryless P Y,Z X Y n Dec M (nr bits) Z n R I(X;Z) Given M, can decode M.

15 Wyner s security argument I(M,M ;Z n ) = I(M;Z n ) + I(M ;Z n M) I(X n ;Z n ) ni(x;z) H(M ) = nr Decodable if R <I(X;Z)

16 Secrecy Metric Secrecy capacity asks for perfect secrecy Lossless compression near-lossless as Perfect secrecy near-perfect

17 Weak Secrecy Wyner s proof establishes weak perfect secrecy I(M;Z n ) can be made arbitrarily small compared to n

18 Strong Secrecy Recent proofs focus on strong perfect secrecy. I(M;Z n ) can be made arbitrarily small Warning: Strong is not strong enough!

19 Modern Proof Use soft-covering principle from Wyner s other 1975 paper

20 Soft Covering M (nr bits) Code u n (M) Memoryless Q V U V n R > I(U;V) Randomly select a codeword Pass through a memoryless channel Output is i.i.d.

21 Output Distribution Desired output distribution: Q V (v)= X u Q V U (v u)q U (u) Q V n Q U n = Y Q V = Y Q U Induced output distribution: P V n C =2 nr X u n (m)2c Q V n U n =u n (m) Q V n U n = Y Q V U

22 Output Distribution P V C Q V u(1) u(5) u(4) u(2) u(3)

23 Soft Covering Lemma Codebook size: R>I(U;V) Codebook generation: U n (m) Q U i.i.d. Success: P V n C Q V n

24 Gaussian Example

25 Gaussian Example

26 Covering and Packing Covering (compression) Packing (transmission) U n V n U n V n

27 Covering Hard covering: [ u n (m) T (u n (m)) V n in probability Soft covering: 1 X 2 nr u n (m)2c Q V n U n =u n (m) Q V n

28 Soft Covering Metrics Wyner: E 1 n D P V n CkQ V n! 0 Han-Verdú resolvability (1993): E P V n C Q V n TV! 0 Many other proofs and uses: E D P V n CkQ V n! 0

29 Back to Wiretap Channel Same encoding construction as previously Use soft covering to show secrecy

30 Encoding Concept M (nr bits) Enc X n Memoryless P Y,Z X Y n Dec M (nr bits) Z n R I(X;Z) Given M, distribution P Z

31 Stronger Soft Covering Lemma

32 Claim P D(P V n CkQ V n) >e 1n <e e 2n Conditions: - R > I(U;V) - For some γ 1 and γ 2 and n large enough - V has finite support

33 Proof Concept Typical part close to 1 for all v n C(v n ), dp V n C dq V n (v n ). Z D P V n CkQ V n = dp V n C log C.

34 Existence argument Performance error metrics: e 1,n, e 2,n, must all go to zero Expected value over codebooks: E [e 1,n +e 1,n ] = E[e 1,n ] + E[e 1,n ] ε Probability of a good codebook: P(e 1,n >ε OR e 1,n >ε ) P(e 1,n >ε) + P(e 1,n >ε)

35 Semantic Security Goldwasser-Micali 1982 No test can distinguish between a random selection from any two messages (P error 1/2) P. Mi - P. Mk TV 0 for all i,k

36 Strong is Too Weak Message is assumed to be uniformly distributed I(M; Z n )=2 nr X m D P Zn M=mkP Z n close on average

37 Example Encoding is in packets of 256 bits End user only needs to use 3/4 of the packet during each transmission By protocol, end user fills the end of the packet with 0 s Can have no security and still strong perfect secrecy! Uniform mutual information (3/4) n

38 Semantic Security as Mutual Information Bellare-Tessaro-Vardy 2012 Equivalence: Semantic security max P M I(M; Z n ) <

39 Expurgation Semantic Security in Wiretap Channel Easy way: Remove bad codewords Another easy way: Use stronger soft-covering lemma

40 More Complex Wiretap Channels Ziv Goldfeld and Haim Permuter

41 Type II Wiretap Channel Ozarow-Wyner 1984 No noise Eavesdropper selects to αn packets to observe out of n transmitted

42 Type II Wiretap Channel Eavesdropper

43 Solution Achieve secrecy capacity of wiretap channel No noise Erasure probability to the eavesdropper (1-α) Use coset codes

44 Noisy Main Channel Nafea-Yener 2015 Built on coset code construction Not optimal in general Goldfeld-Cuff-Permuter 2015 Achieve wiretap channel secrecy capacity in general Semantic security capacity is the same

45 Arbitrarily Varying Wiretap Channel M (nr bits) Enc X n Memoryless P Y,Z X,S Y n Dec Z n Secrecy Capacity: S n - Reliable communication - Z n contains no information about M

46 Secrecy Proof 1. Analyze random codebook 2. Consider an arbitrary message and eavesdropper choice Exponential number of these 3. Soft-covering lemma 4. Union bound

47 Wiretap Channel with State M (nr bits) Enc X n Memoryless P Y,Z X,S Y n Dec Z n S n Secrecy Capacity: - Reliable communication - Z n contains no information about M

48 Highlights Breakthroughs for Physical-layer Security Strong Soft Covering Semantic Security Wiretap Channels with State

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