RELIABLE COMMUNICATION 1 IN THE PRESENCE OF LIMITED ADVERSARIES Sidharth Jaggi The Chinese University of Hong Kong Between Shannon and Hamming Codes against limited adversaries 3 Qiaosheng Zhang ID: 763148
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RELIABLE COMMUNICATION 1 IN THE PRESENCE OF LIMITED ADVERSARIES Sidharth Jaggi , The Chinese University of Hong Kong
Between Shannon and Hamming: Codes against limited adversaries
3 Qiaosheng ZhangMayank BakshiSidharth Jaggi Swanand KadheAlex Sprintson Michael Langberg Zitan Chen ME Anand Dilip Sarwate Bikash Kumar Dey Research group: Codes, Algorithms, Networks: Design & Optimization for Information Theory C odes, A lgorithms, N etworks: D esign & O ptimization for I nformation T heory
Background – Communication Scenario 4 Alice Bob 011100110110111101110011 Calvin Bad guy Encoder Decoder (Adversarial) Noisy Channel M essage D ecoded message Codeword Received word
Background – Related Work 5 “Benchmark” channel models One extreme: random noise (Shannon) errors Alphabet q=2 (binary ) R p 1 0.5 “ large ” q R p 1 1 {Sha48] [Sha48] ( q -ary symmetric channel) = u k w.h.p . general q R p 1 1-1/q [Sha48] Many “good” codes (computationally efficient, “close to” capacity)
Background – Related Work 6 “Benchmark” channel models The other extreme: omniscient adversary (Hamming) One extreme: random noise (Shannon) q=2 (binary ) R p 1 0.5 “ large ” q R p 1 1 [Reed-Solomon ]/[Singleton] [McERRW77] [ Gilbert Varshamov ] Calvin He knows “everything”! “Good, computationally efficient” codes for large q , not so much for small q 0.25 0.5 1-2p
Avg P e : Max P e : x 1 (0), p 1 x 2 (0), p 2 x 3 (0), p 3 x 4 (0), p 4 x 5 (0), p 5 R p 1 0.5
Avg P e : Max P e : Max P e : R p 1 0.5 [McERRW77] [ Gilbert Varshamov ] R p 1 0.5 Avg P e :
Background – Related Work 9 “Benchmark” channel models The other extreme: omniscient adversary One extreme: random noise q=2 (binary ) R p 1 0.5 “ large ” q R p 1 1 [Sha48] [Reed-Solomon ]/[Singleton] [McERRW77] [ Gilbert Varshamov ] Calvin He knows “everything”! “ intermediate ” q p 1 0.5
Background – Related Work 10 “Benchmark” channel models One intermediate model: oblivious adversary errors Alphabet q=2 (binary ) R p 1 0.5 “ large ” q R p 1 1 (AVC capacity) (AVC capacity/folklore)
Background – Related Work 11 “Benchmark” channel models One intermediate model: oblivious adversary errors Alphabet q=2 (binary ) R p 1 0.5 “ large ” q R p 1 1 (AVC capacity) (AVC capacity/folklore) s s : Secret key known only to Alice (NOT Bob/Calvin) e n : Error function of codebook, message u k , but NOT codeword x n “Good, computationally efficient” codes recently constructed for q =2 [Guruswami-Smith-10]
Background – Related Work 12 “Benchmark” channel models Another intermediate model: common-randomness between Alice/Bob errors Alphabet q=2 (binary ) R p 1 0.5 “ large ” q R p 1 1 (AVC capacity) (AVC capacity/folklore) s s : Secret key known to Alice and Bob (NOT Calvin) e n : Error function of codebook, codeword x n , but NOT u k û k : decoded codeword function of s “Good, computationally efficient” codes ( Eg : Ahlswede’s permutation trick.) , s
Background – Related Work 13 Weaker channel models List-decoding - weakened reconstruction goal q=2 (binary ) R p 1 0.5 “ large ” q R p 1 1 [ Elias,Wozencfraft ] [ Elias,Wozencfraft ]
Background – Related Work 14 Weaker channel models List-decoding - weakened reconstruction goal q=2 (binary ) R p 1 0.5 “ large ” q R p 1 1 [ Elias,Wozencfraft ] [ Elias,Wozencfraft ] “Good, computationally efficient” codes recently constructed for large q [Guruswami-Rudra06], q =2 open
Background – Related Work 15 Weaker channel models List-decoding - weakened reconstruction goal q=2 (binary list-decoding) R p 1 0.5 “ large ” q R p 1 1 Smith- Guruswami Computationally efficient encoding/decoding schemes Computationally bounded adversaries - weakened adversary power Calvin Micali -Sudan (modified with Guruswami-Rudra )
Background – Related Work 16 Weaker channel models AVCs with common randomness between Alice and Bob Omniscient jammer with noiseless feedback Calvin Alice Bob q=2 (binary ) R p 1 0.5 [Sha48] [McERRW77] [ Gilbert Varshamov ] [ Berlekamp ] 1/3 Only deals with -error ( -error capacity open ) “ large ” q R p 1 0.5
Calvin 18 Causal adversaries Between oblivious and omniscient adversaries 0 1 1 ? 1 0 ? ? ? ? ? ? ? 1 3 2 10 4 5 6 7 8 9 13 11 12 0 0 1 1 * ? 14 Transmitted Word Tampered Word Current Future Calvin Calvin Calvin Calvin
19 Causal adversaries Between oblivious and omniscient adversariesCausal large alphabetDelayed adversary Causal “ large ” q R p 1 0.5 Delayed q=2 R p 1 0.5 Delayed “ large ” q (additive) R p 1 1 R p 1 1 d 0.5 Delayed “ large ” q (overwrite) “ Sha-mming ”
20 Causal adversaries Capacity “ Sha-mming ” (Chen, J, Langberg , STOC 2015)
21 Causal adversaries Between oblivious and omniscient adversaries Analysis of all possible causal adversarial behaviours 1 One possible adversarial trajectory (Slopes are bounded)
Analysis of all possible causal adversarial behaviours Proof techniques overview - Converse: “Babble-and-push” attack 22 Causal adversaries 0 1 1 ? 1 0 ? ? ? ? ? ? ? 1 3 2 10 4 5 6 7 8 9 13 11 12 0 0 1 1 1 ? 14 Transmitted Word Tampered Word Babbling phase Pushing phase Randomly tamper with bits
Proof techniques overview - Converse : “Babble-and-push” attack 23 Causal adversaries Transmitted Word Tampered Word 1 … 0 0 1 0 1 1 0 10 … 6 7 8 9 13 11 12 1 … 0 1 1 0 1 0 0 1 14 1 Selected Word 1 … 0 1 0 1 1 0 0 1 Pushing phase Construct a set of codewords based on corrupted bits transmitted so far Select one codeword from the set and then “push” the transmitted codeword towards the selected one Pushing phase
Proof techniques overview - Converse : “Babble-and-push” attack 24 Causal adversaries Transmitted Word Tampered Word 1 … 0 0 1 0 1 1 0 10 … 6 7 8 9 13 11 12 1 … 0 1 1 0 1 0 0 1 14 1 Selected Word 1 … 0 1 0 1 1 0 0 1 Pushing phase Construct a set of codewords based on corrupted bits transmitted so far Select one codeword from the set and then “push” the transmitted codeword towards the selected one Pushing phase The tampered word lies in midway between the transmitted word and selected word .
Proof techniques overview - Converse : “Babble-and-push” attack 25 Causal adversaries Group of size Plotkin bound: Binary code with d min > n(1+ε)/2 has O(1/ε) codewords
26 Causal adversaries 0 1 1 ? 1 0 ? ? ? ? ? ? ? 1 3 2 10 4 5 6 7 8 9 13 11 12 0 0 1 1 1 ? 14 Transmitted Word Tampered Word List-decoding condition Proof techniques overview - Converse : “Babble-and-push” attack Energy-bounding condition Babbling phase Pushing phase # message bits Shannon-capacity of first l channel uses remaining bit-flip budget Avg pairwise distance ( Plotkin )
Proof techniques overview - Converse : “Babble-and-push” attack 27 Causal adversaries Using stochastic encoders instead of deterministic encoders illustrated in previous slides Message
Proof techniques overview - Converse : “Babble-and-push” attackCalvin won’t run out of his flipping budgetImposing a probability of decoding error 28 Causal adversaries Calvin
Proof techniques overview - Converse : “Babble-and-push” attack 29 Causal adversaries Proof techniques overview – Achievability 1 Trajectory of the “babble-and-push strategy” Possible decoding points
Proof techniques overview - Converse : “Babble-and-push” attack 30 Causal adversaries Proof techniques overview – Achievability Encoder: concatenated stochastic codes code bits bits bits
31 Causal adversaries Proof techniques overview – AchievabilityEncoder: concatenated stochastic codes code bits bits bits bits
32 Causal adversaries Proof techniques overview – AchievabilityEncoder: concatenated stochastic codesDecoding process: list-decoding + unique decoding bits Obtain a list of messages List-decoding phase Unique decoding phase
33 Causal adversaries Proof techniques overview – AchievabilityEncoder: concatenated stochastic codesDecoding process: list-decoding + unique decoding bits Obtain a list of messages List-decoding phase Unique decoding phase Encodings Consistency Checking If two words differ in a limited number of positions, they are said to be consistent.
34 Causal adversaries Proof techniques overview – AchievabilityEncoder: concatenated stochastic codesDecoding process: list-decoding + unique decoding List of “right mega sub- codewords ” Fails consistency checking… Received “right mega sub- codeword ”
35 Causal adversaries Proof techniques overview – AchievabilityEncoder: concatenated stochastic codesDecoding process: list-decoding + unique decoding With high probability, Bob succeeds in decoding Passes consistency checking! Received “right mega sub- codeword ”
36 Summary (Chen, J, Langberg STOC 2015) Binary Bit-flips Binary Erasures
37 Sneak Previews (Chen, Jaggi, Langberg, ISIT2016) q-ary online p*-erasure p -error channels ( Dey , Jaggi , Langberg , Sarwate ISIT2016) one-bit delay p *-erasure channels Stochastic encoding Deterministic encoding
0101001011001011001011000101000101001010110 0101001011001011001011000101000101001010110 0101001011001011001011000101000101001010110 . . . Coherence.pptx Coherence.pptx Coherence.pptx
q1 q2 q3 q4 q5 q5 q4 q3 q2 q1 q1 q2 q3 q4 q5 q5 q4 q3 q2 q1 q1 q2 q3 q4 q5q2 q4 q 5 q1 q3 q1 q 2 q3 q 4 q5 q 3 q 2 q 1 q 5 q 4 q 5 q 2 q 3 q 1 q 2 q 3 q 5 q 2 q 3 q1
1 0 0 1 0 1 0 1 1 Erasures Erasing branch-points Erasing disambiguity
41 Limited-view adversaries: Multipath networks with large-alphabet symbols Adversary can see a certain fraction and jam another fractionMyopic adversaries: Adversary has (non-causal) view of a noisy version of Alice's transmission ITW 2015,ISIT 2015 ISIT 2015 Other related models
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