RELIABLE COMMUNICATION IN THE PRESENCE OF LIMITED ADVERSARIES

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2 2 ELIABLE COMMUNICATION IN THE PESENCE OF LIMITED ADVESAIES

3 3 ME esearch grou: Codes, Codes, Algorithms, Algorithms, Networks: Networks: Design & Otimization for Information Information Theory Theory Zitan Chen Qiaosheng Zhang Mayank Bakshi Sidharth Jaggi Swanand Kadhe Alex Srintson Michael Langberg Bikash Kumar Dey Anand Dili Sarwate

4 Background Communication Scenario 4 Bad guy Alice Calvin Bob u k x n y n u k Message Encoder Codeword (Adversarial) Noisy Channel eceived word Decoder Decoded message

5 Background elated Work 5 Benchmark channel models One extreme: random noise (Shannon) u k x n y n u k = u k w.h.. Alhabet q = k n e n n errors (q-ary symmetric channel) q=2 (binary) general q large q {Sha48] [Sha48] [Sha48] 0.5 -/q Many good codes (comutationally efficient, close to caacity)

6 Background elated Work 6 Benchmark channel models The One other extreme: extreme: random omniscient noise (Shannon) adversary (Hamming) u k x n y n u k Calvin He knows everything! q=2 (binary) large q -2 [Gilbert Varshamov] [McEW77] [eed- Solomon]/[Singleton] Good, comutationally efficient codes for large q, not so much for small q

7 x 2 (0), 2 x (0), x 3 (0), 3 x 4 (0), 4 x 5 (0), 5 Avg P e : Max P e : 0.5

8 Avg P e : Avg P e : Max P e : Max P e : 0.5 [Gilbert Varshamov] [McEW77] 0.5

9 Background elated Work 9 Benchmark channel models One intermediate model: oblivious adversary u k x n y n u k s s: Secret key known only to Alice (NOT Bob/Calvin) Alhabet q = k n e n n errors e n : Error function of codebook, message u k, but NOT codeword x n q=2 (binary) large q (AVC caacity) (AVC caacity/folklore) 0.5 Good, comutationally efficient codes recently constructed for q=2 [Guruswami-Smith-0]

10 Background elated Work 0 Benchmark channel models Another intermediate model: common-randomness between Alice/Bob s: Secret key known to Alice and Bob (NOT Calvin) u k x n y n u k s Alhabet q = k n q=2 (binary) e n n errors large q, e n : Error function of codebook, codeword x n, but NOT u k û k : decoded codeword function of s s (AVC caacity) (AVC caacity/folklore) 0.5 Good, comutationally efficient codes (Eg: Ahlswede s ermutation trick.)

11 Background elated Work Weaker channel models List-decoding - weakened reconstruction goal u k x n y n q=2 (binary) large q [Elias,Wozencfraft] [Elias,Wozencfraft] 0.5 Good, comutationally efficient codes recently constructed for large q [Guruswami-udra-06], q=2 oen

12 Background elated Work 2 Weaker channel models List-decoding Comutationally - weakened bounded reconstruction adversaries - weakened goal adversary ower u k x n y n Calvin u k Comutationally efficient encoding/decoding schemes q=2 (binary) large q Smith-Guruswami Micali-Sudan 0.5

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14 4 Between oblivious and omniscient adversaries Current Future Transmitted Word Tamered Word ???????? 0 0 * 4? Calvin Calvin Calvin Calvin Calvin

15 5 Between oblivious and omniscient adversaries Causal large alhabet Delayed adversary Causal large q Delayed q=2 Delayed large q (additive) Delayed large q (overwrite) d Sha-mming

16 6 Caacity STOC 205 Sha-mming

17 7 Between Analysis of oblivious all ossible and causal omniscient adversarial adversaries behaviours t One ossible adversarial trajectory (Sloes are bounded) t n

18 8 Analysis Proof techniques of all ossible overview causal - Converse: adversarial Babble-and-ush behaviours attack Transmitted Word Tamered Word ???????? 0 0 4? Babbling hase Pushing hase andomly tamer with n bits

19 9 Proof techniques overview - Converse: Babble-and-ush attack Pushing hase. Construct a set of codewords based on corruted bits transmitted so far 2. Select one codeword from the set and then ush the transmitted codeword towards the selected one Transmitted Word Tamered Word Selected Word Pushing hase 4

20 20 Proof techniques overview - Converse: Babble-and-ush attack Pushing hase. Construct a set of codewords based on corruted bits transmitted so far 2. Select one codeword from the set and then ush the transmitted codeword towards the selected one Transmitted Word Tamered Word Selected Word Pushing hase 4 The tamered word lies in midway between the transmitted word and selected word.

21 2 Proof techniques overview - Converse: Babble-and-ush attack Plotkin bound: Binary code with d min > n(+ε)/2 has O(/ε) codewords Grou of size ε

22 22 Proof techniques overview - Converse: Babble-and-ush attack Babbling hase Pushing hase Transmitted Word Tamered Word ???????? 0 0 4? l List-decoding condition n l Energy-bounding condition Shannon-caacity of first l channel uses # message bits remaining bit-fli budget Avg airwise distance (Plotkin)

23 23 Proof techniques overview - Converse: Achievability Babble-and-ush attack Possible decoding oints t Trajectory of the babble-and-ush strategy t n

24 24 Proof techniques overview - Converse: Achievability Babble-and-ush attack Encoder: concatenated stochastic codes m s code C C (m, s ) n bits ns bits nθ bits

25 25 Proof techniques overview Achievability Encoder: concatenated stochastic codes m s code C C (m, s ) C (m, s ) C 2 (m, s 2 ) C θ (m, s θ ) n bits ns bits n bits nθ bits

26 26 Proof techniques overview Achievability Encoder: concatenated stochastic codes Decoding rocess: list-decoding + unique decoding C (m, s ) C 2 (m, s 2 ) C t (m, s t ) C θ (m, s θ ) tnθ bits List-decoding hase Unique decoding hase Obtain a list of messages

27 27 Proof techniques overview Achievability Encoder: concatenated stochastic codes Decoding rocess: list-decoding + unique decoding If two words differ in a limited number of ositions, they are said to be consistent. C (m, s ) C 2 (m, s 2 ) C t (m, s t ) C θ (m, s θ ) Obtain a list of messages tnθ bits List-decoding hase Encodings Unique decoding hase Consistency Checking

28 28 Proof techniques overview Achievability Encoder: concatenated stochastic codes Decoding rocess: list-decoding + unique decoding List of right mega sub-codewords eceived right mega sub-codeword Fails consistency checking

29 29 Proof techniques overview Achievability Encoder: concatenated stochastic codes Decoding rocess: list-decoding + unique decoding With high robability, Bob succeeds in decoding eceived right mega sub-codeword Passes consistency checking!

30 30 Limited-view adversaries: Multiath networks with large-alhabet symbols Adversary can see a certain fraction and jam another fraction ITW 205, ISIT 205 Myoic adversaries: Adversary has (non-causal) view of a noisy version of Alice's transmission ISIT 205

31 Questions?