Joint Source-Channel Secrecy Using Hybrid Coding Eva Song, Paul - PowerPoint PPT Presentation

Joint Source-Channel Secrecy Using Hybrid Coding Eva Song, Paul Cuff, and H. Vincent Poor Department of Electrical Engineering Princeton University June 19, 2015

A source-channel coding setting t = 1 , . . . , n ˆ S t Decoder g n Y n S n X n P YZ | X Encoder f n ˇ Z n S t Eve S t − 1 Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 2 / 22

A source-channel coding setting t = 1 , . . . , n ˆ S t Decoder g n Y n S n X n P YZ | X Encoder f n ˇ Z n S t Eve S t − 1 Quality of reconstruction: d ( S n , ˆ S n ), d ( S n , ˇ S n ) Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 2 / 22

A source-channel coding setting t = 1 , . . . , n ˆ S t Decoder g n Y n S n X n P YZ | X Encoder f n ˇ Z n S t Eve S t − 1 Quality of reconstruction: d ( S n , ˆ S n ), d ( S n , ˇ S n ) Why causal disclosure? ◮ Stronger formulation: to the favor of eavesdropper ◮ Can generalize equivocation Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 2 / 22

In this talk... Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 3 / 22

In this talk... Design source-channel coding schemes for ( D b , D e ) s.t. � � d ( S n , ˆ ◮ E S n ) ≤ n D b t =1 E [ d ( S n , ˇ ◮ min { P ˇ S n )] ≥ n D e St | ZnSt − 1 } n Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 3 / 22

In this talk... Design source-channel coding schemes for ( D b , D e ) s.t. � � d ( S n , ˆ ◮ E S n ) ≤ n D b t =1 E [ d ( S n , ˇ ◮ min { P ˇ S n )] ≥ n D e St | ZnSt − 1 } n Analysis uses The Likelihood Encoder ◮ Total variation distance ◮ Soft-covering lemma Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 3 / 22

What is a likelihood encoder? a stochastic source encoder: f n : X n �→ M X n Y n M Decoder g n Encoder f n Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 4 / 22

What is a likelihood encoder? a stochastic source encoder: f n : X n �→ M X n Y n M Decoder g n Encoder f n Given a codebook { y n ( m ) } m , m ∈ [1 : 2 nR ] a joint distribution P XY the likelihood function for each codeword: L ( m | x n ) � P X n | Y n ( x n | y n ( m )) = � P X | Y ( x n | y n ( m )) the likelihood encoder determines the message index according to: L ( m | x n ) P M | X n ( m | x n ) = m ′ ∈ [1:2 nR ] L ( m ′ | x n ) ∝ L ( m | x n ) . � Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 4 / 22

Warm up – soft-covering lemma Lemma Given 1) P UXZ 2) random C ( n ) of sequences U n ( m ) ∼ � n t =1 P U ( u t ) , m ∈ [1 : 2 nR ] Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 5 / 22

Warm up – soft-covering lemma Lemma Given 1) P UXZ 2) random C ( n ) of sequences U n ( m ) ∼ � n t =1 P U ( u t ) , m ∈ [1 : 2 nR ] Let n k 1 � � P MX n Z k ( m , x n , z k ) � P X | U ( x t | U t ( m )) P Z | XU ( z t | x t , U t ( m )) 2 nR t =1 t =1 n k � � P X n Z k � P X ( x t ) P Z | X ( z t | x t ) t =1 t =1 Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 5 / 22

Warm up – soft-covering lemma Lemma Given 1) P UXZ 2) random C ( n ) of sequences U n ( m ) ∼ � n t =1 P U ( u t ) , m ∈ [1 : 2 nR ] Let n k 1 � � P MX n Z k ( m , x n , z k ) � P X | U ( x t | U t ( m )) P Z | XU ( z t | x t , U t ( m )) 2 nR t =1 t =1 n k � � P X n Z k � P X ( x t ) P Z | X ( z t | x t ) t =1 t =1 If R > I ( X ; U ) , then E C n �� � � � P X n Z k − P X n Z k ≤ exp( − γ n ) → n 0 , � TV for any β < R − I ( X ; U ) I ( Z ; U | X ) , k ≤ β n, γ > 0 depending on this gap. Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 5 / 22

Problem setup t = 1 , . . . , n ˆ S t Decoder g n Y n S n X n P YZ | X Encoder f n ˇ Z n S t Eve S t − 1 i.i.d. source S n ∼ � n t =1 P S ( s t ) memoryless broadcast channel � n t =1 P YZ | X ( y t , z t | x t ) Encoder f n : S n �→ X n (possibly stochastic) Legitimate receiver decoder g n : Y n �→ ˆ S n (possibly stochastic) S t | Z n S t − 1 } n Eavesdropper decoders { P ˇ t =1 Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 6 / 22

Definition Definition A distortion pair ( D b , D e ) is achievable if there exists a sequence of source-channel encoders and decoders ( f n , g n ) such that E [ d ( S n , ˆ S n )] ≤ n D b and E [ d ( S n , ˇ S n )] ≥ n D e . min St | ZnSt − 1 } n { P ˇ t =1 Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 7 / 22

We consider Scheme O – Operationally separate SC coding [Schieler et al. Allerton 2012] Scheme I – Joint SC coding using Hybrid Coding Scheme II – Joint SC coding using superposition Hybrid Coding Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 8 / 22

Scheme O – operational separate Theorem A distortion pair ( D b , D e ) is achievable if I ( S ; U 1 ) < I ( U 2 ; Y ) I ( S ; ˆ S | U 1 ) < I ( V 2 ; Y | U 2 ) − I ( V 2 ; Z | U 2 ) � � d ( S , ˆ D b ≥ E S ) D e ≤ η min E [ d ( S , a )] + (1 − η ) min t ( u 1 ) E [ d ( S , t ( U 1 ))] a ∈ ˆ S for some distribution P S P ˆ S | S P U 1 | ˆ S P U 2 P V 2 | U 2 P X | V 2 P YZ | X , where η = [ I ( U 2 ; Y ) − I ( U 2 ; Z )] + . I ( S ; U 1 ) Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 9 / 22

Hybrid coding U n ( ˆ U n ( M ) M ) ˆ S n X n Y n S n P X | SU P YZ | X φ ( u , y ) Likelihood Encoder Channel Decoder at least optimal for P2P communication [Minero et al.] achieves best known bounds in multiuser settings Secrecy: need stochastic symbol-by-symbol mapping Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 10 / 22

Scheme I – basic hybrid coding Theorem A distortion pair ( D b , D e ) is achievable if I ( U ; S ) I ( U ; Y ) < D b ≥ E [ d ( S , φ ( U , Y ))] D e ≤ β min ψ 0 ( z ) E [ d ( S , ψ 0 ( Z ))] +(1 − β ) min ψ 1 ( u , z ) E [ d ( S , ψ 1 ( U , Z ))] where � [ I ( U ; Y ) − I ( U ; Z )] + � β = min , 1 I ( S ; U | Z ) for some distribution P S P U | S P X | SU P YZ | X and function φ ( · , · ) . Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 11 / 22

Scheme I – achievability scheme Fix distribution P S P U | S P X | SU P YZ | X Codebook generation: Independently generate 2 nR sequences in U n according to � n t =1 P U ( u t ) and index by m ∈ [1 : 2 nR ] Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 12 / 22

✶ Scheme I – achievability scheme – continued Encoder ◮ likelihood encoder P LE ( m | s n ) with L ( m | s n ) = P S n | U n ( s n | u n ( m )) ◮ produces channel input through a random transformation: � n t =1 P X | SU ( x t | s t , U t ( m )) Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 13 / 22

Scheme I – achievability scheme – continued Encoder ◮ likelihood encoder P LE ( m | s n ) with L ( m | s n ) = P S n | U n ( s n | u n ( m )) ◮ produces channel input through a random transformation: � n t =1 P X | SU ( x t | s t , U t ( m )) Decoder ◮ good channel decoder P D 1 ( ˆ m | y n ) w.r.t. codebook { u n ( a ) } a and memoryless channel P Y | U ◮ deterministic mapping φ n ( u n , y n ) is the concatenation of { φ ( u t , y t ) } n t =1 : s n = φ n ( u n ( ˆ s n | ˆ m , y n ) � ✶ { ˆ m ) , y n ) } P D 2 (ˆ Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 13 / 22

Analysis outline – at legitimate receiver System induced distribution P Idealized distribution Q Q MU n S n X n Y n Z n ( m , u n , s n , x n , y n , z n ) n 1 2 nR ✶ { u n = U n ( m ) } � � P S | U ( s t | u t ) t =1 n n � � P X | SU ( x t | s t , u t ) P YZ | X ( y t , z t | x t ) . t =1 t =1 soft-covering: R > I ( U ; S ) ⇒ P ≈ Q channel coding: R ≤ I ( U ; Y ) ⇒ � � �� d ( S n , ˆ S n ) E C ( n ) E P ≤ E P [ d ( S , φ ( U , Y ))] + δ n Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 14 / 22

Analysis outline – at eavesdropper auxiliary distribution n i Q ( i ) ˇ � � S i Z n ( s i , z n ) � P Z ( z t ) P S | Z ( s j | z j ) t =1 j =1 Q ( i ) soft-covering: R > I ( Z ; U ) ⇒ ˇ Z n S i ≈ Q Z n S i i can go up to β n , for any β < R − I ( U ; Z ) I ( S ; U | Z ) phase transition in distortion ◮ before β n : � � � k ◮ min { ψ 0 i ( s i − 1 , z n ) } i E P 1 i =1 d ( S i , ψ 0 i ( S i − 1 , Z n )) ≥ k min ψ 0 ( z ) E P [ d ( S , ψ 0 ( Z ))] − ǫ n ◮ after β n : � � � n ◮ min { ψ 1 i ( s i − 1 , z n ) } E P 1 i = j d ( S i , ψ 1 i ( S i − 1 , Z n )) ≥ k min ψ 1 ( u , z ) E P [ d ( S , ψ 1 ( U , Z ))] − ǫ n Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 15 / 22