On Secure Asymmetric Multilevel Diversity Coding Systems Congduan Li Sun Yat-sen University Jun 21-26, 2020 ISIT 2020 (LA, USA, online) Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 1 / 27
Outline 1 Motivation 2 System Model 3 An example 4 Extension to general case Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 2 / 27
Outline 1 Motivation 2 System Model 3 An example 4 Extension to general case Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 3 / 27
Related work Related work in TransIT [Jiang et al. 2014] Wiretap channel on symmetric multilevel diversity coding systems (SMDCS) Secure rate region obtained Superposition (source separation) coding optimal Natural question: what about asymmetric case? This work: secure rate region for asymmetric MDCS (AMDCS) and optimality of achieving coding schemes Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 4 / 27
Outline 1 Motivation 2 System Model 3 An example 4 Extension to general case Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 5 / 27
Secure AMDCS Build on the system model of AMDCS by [Mohajer et al. 2010] Impose the secrecy constraints as in [Jiang et al. 2014] Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 6 / 27
Secure AMDCS Independent sources are prioritized: X 1 is more important than X 2 , X 2 is more important than X 3 , and so on. More important sources must be decoded before decoding less important sources. Secrecy key independent to sources Secrecy Encoders Key and Decoders Prioritized X 1 Fan( D 1 ) D 1 Sources U 1 E 1 K X 1 Paths D i X 1: i X 2 K, X 1: S . . . X S D S E L U L X 1: S Fan( D S ) Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 7 / 27
Secure AMDCS Sources, together with secrecy key, are coded by multiple encoders Eavesdropper may has access to one of the size m (security level) subset of encoders (wiretap channel II), but can know nothing about the sources Secrecy Eavesdropper Encoders Key and Decoders Size m subset Prioritized Fan( D 1 ) X 1 D 1 Sources U 1 E 1 K X 1 Paths D i X 1: i X 2 K, X 1: S . . . X S D S E L U L X 1: S Fan( D S ) Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 8 / 27
Secure AMDCS Input to decoder ( Fan ( D i )) should include at least m + 1 encoders Symmetric case: all decoders with input of m + i encoders can decode X 1 , . . . , X i General case: each secure transmission has an associated decoder, and decoder D i can decode X 1 , . . . , X i Order: Naturally, a decoder whose input is a subset of another decoder’s input, its decoding level should be smaller Secrecy Eavesdropper Key and Encoders Decoders Size m subset Prioritized Fan( D 1 ) X 1 D 1 Sources U 1 E 1 K X 1 Paths D i X 1: i X 2 K, X 1: S . . . X S D S E L U L X 1: S Fan( D S ) Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 9 / 27
Secure Rate region Secure rate region: all possible rate and source entropy vectors satisfying all network constraints and secrecy constraints, with existence of proper secrecy key, coding and decoding functions. Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 10 / 27
Outline 1 Motivation 2 System Model 3 An example 4 Extension to general case Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 11 / 27
AMDCS with 4 encoders and security level 2 4 encoders, 5 sources, 5 decoders Each decoder has a distinct X 1 D 1 decoding level K E 1 X 1 , X 2 X 1 Order: D 2 E 2 X 2 L 1 ( E 1 E 2 E 3 ) = 1 X 1 , X 2 , X 3 D 3 X 3 E 3 L 1 ( E 1 E 2 E 4 ) = 2 X 1 , X 2 , X 3 , X 4 D 4 X 4 L 1 ( E 1 E 3 E 4 ) = 3 X 1 , X 2 , X 3 , X 4 , X 5 X 5 E 4 L 1 ( E 2 E 3 E 4 ) = 4 D 5 security : L 1 ( E 1 E 2 E 3 E 4 ) = 5 I ( X 1:5 ; E i,j ) = 0 , i, j ∈ { 1 , 2 , 3 , 4 } First obtain the inner bound, superposition secure rate region Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 12 / 27
Superposition: sources coded separately r 1 X 1 , K i } Each source, with key, is coded separately X 2 , K r 2 i Coding rate is sum of sub-rates on r 3 R i X 3 , K i each source: R i = � 5 j =1 r j i , i = 1 , . . . , 5 r 4 X 4 , K i For each sub-encoder, it is a solved r 5 single-source secrecy problem [Shamir X 5 , K i 1979, Cai-Yeung 2011] E i Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 13 / 27
Superposition secure rate region From decoder X 1 D 1 K D i , i = 1 , . . . , 4, we have E 1 X 1 , X 2 r j X 1 D 2 k ≥ H ( X j ), E 2 X 2 X 1 , X 2 , X 3 k ∈ Fan ( D i ) , j = 1 , . . . , i D 3 X 3 E 3 From decoder D 5 , we X 1 , X 2 , X 3 , X 4 D 4 X 4 have r j i + r j k ≥ H ( X j ), X 5 E 4 X 1 , X 2 , X 3 , X 4 , X 5 D 5 i , k ∈ Fan ( D 5 ) , i � = k , j = security : I ( X 1:5 ; E i,j ) = 0 , i, j ∈ { 1 , 2 , 3 , 4 } 1 , . . . , 5 Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 14 / 27
Superposition secure rate region 3 � R 1 H ( X i ) , ≥ Together with the rate i =1 sum equations and 4 � eliminate sub-rates, we R j H ( X i ) , j = 2 , 3 , 4 ≥ i =1 get the inner bound 3 5 � � The achievability can be R 1 + R i 2 H ( X j ) + H ( X l ) , i = 2 , 3 , 4 ≥ verified by constructing j =1 l =4 4 the codes for each � R 2 + R i 2 H ( X j ) + H ( X 5 ) , i = 3 , 4 ≥ extreme ray of the j =1 polyhedra formed by the 4 � R 3 + R 4 2 H ( X i ) + H ( X 5 ) ≥ inequalities i =1 Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 15 / 27
Exact secure rate region If coding across sources are allowed, the secure rate region is enlarged Some coefficients are changed Superposition Exact 3 3 � � H ( X i ) , H ( X i ) R 1 ≥ R 1 ≥ i =1 i =1 4 4 � � H ( X i ) , j = 2 , 3 , 4 H ( X i ) , j = 2 , 3 , 4 R j ≥ R j ≥ i =1 i =1 3 5 3 5 � � � � R 1 + R i 2 H ( X j ) + H ( X l ) , i = 2 , 3 , 4 R 1 + R j 2 H ( X i ) + H ( X k ) , j = 2 , 3 , 4 ≥ ≥ j =1 i =1 l =4 k =4 3 5 4 � � � R 2 + R j 2 H ( X i ) + H ( X k ) , j = 3 , 4 R 2 + R i 2 H ( X j ) + H ( X 5 ) , i = 3 , 4 ≥ ≥ j =1 i =1 k =4 2 4 � R 3 + R 4 2 H ( X i ) + H ( X 3 ) + H ( X 4 ) + H ( X 5 ) � ≥ R 3 + R 4 2 H ( X i ) + H ( X 5 ) ≥ i =1 i =1 Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 16 / 27
Converse: exact secure rate region Different inequalities proved by decoding and secure constraints from different subsets of decoders D 3 3 � R 1 ≥ H ( X i ) X 1 D 1 i =1 K E 1 4 X 1 , X 2 � D 4 H ( X i ) , j = 2 , 3 , 4 R j ≥ X 1 D 2 i =1 E 2 X 2 D 3 , D 4 , D 5 X 1 , X 2 , X 3 3 5 D 3 � � R 1 + R j 2 H ( X i ) + H ( X k ) , j = 2 , 3 , 4 ≥ } X 3 E 3 i =1 k =4 X 1 , X 2 , X 3 , X 4 D 4 3 5 X 4 � � R 2 + R j 2 H ( X i ) + H ( X k ) , j = 3 , 4 ≥ X 5 E 4 X 1 , X 2 , X 3 , X 4 , X 5 i =1 k =4 D 5 2 � R 3 + R 4 ≥ 2 H ( X i ) + H ( X 3 ) + H ( X 4 ) + H ( X 5 ) D 2 , D 4 , D 5 i =1 Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 17 / 27
Achievability: exact secure rate region Need to figure out the new extreme rays when enlarge the rate region New extreme rays are outside the superposition coding region Coding across sources are necessary Construct codes to achieve them It turns out that linear codes are optimal Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 18 / 27
Key constraints Necessary constraints on the size of the key One can define the secure rate region to include the key size, then, the extreme rays will have the necessary key size for the codes Shearer’s Lemma (or Han’s inequality) is used in the proofs 3 � H ( K ) 2 H ( X i ) + H ( X 4 ) + H ( X 5 ) ≥ i =1 4 � H ( K ) 2 H ( X i ) , ≥ i =1 3 5 � � 2 R i + H ( K ) 4 H ( X j ) + 2 H ( X k ) , i = 1 , 2 ≥ j =1 k =4 2 5 � � 2 R i + H ( K ) 4 H ( X j ) + 3 H ( X 3 ) + 2 H ( X k ) , i = 3 , 4 ≥ j =1 k =4 Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 19 / 27
Outline 1 Motivation 2 System Model 3 An example 4 Extension to general case Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 20 / 27
General L encoders with security level L − 2 Secure transmissions include accesses to all size L − 1 subsets of encoders Decoders and the whole set of the encoders X 1 D 1 { E 1 ,...,E L } \ E L L + 1 sources and decoders, each decoder has a distinct decoding level { E 1 , . . . , E L } \ E L − i +1 from 1 to L + 1 D i X 1: i For D i , i = 1 , . . . , L , we have Fan ( D i ) = { E 1 , . . . , E L } \ { E L − i +1 } D L +1 { E 1 , . . . , E L } X 1: L +1 For D L +1 , we have Fan ( D L +1 ) = { E 1 , . . . , E L } Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 21 / 27
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