Reliable Deniable Communication: Hiding Messages in Noise Pak Hou Che Mayank Bakshi Sidharth Jaggi The Chinese University of Hong Kong The Institute of Network Coding

Alice Bob Reliability

Alice Bob Reliability Deniability Willie (the Warden)

Alice’s Encoder M 𝐼𝑓 𝐓=0, 𝐗 = 𝟎 𝐼𝑓 𝐓=1, 𝐗 =𝐸𝑛𝑐(𝐌) 𝐗 T 𝑚𝑒𝑠𝑠𝑎𝑔𝑒 𝐌∈{1, …, 𝑁} t𝑟𝑎𝑛𝑠. 𝑠𝑡𝑎𝑡𝑢𝑠 𝐓∈{0, 1} 𝑁= 2 𝜃( 𝑛 ) 𝑇ℎ𝑟𝑜𝑢𝑔ℎ𝑝𝑢𝑡 𝜏= log 𝑁 𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑡ℎ𝑟𝑜𝑢𝑔ℎ𝑝𝑢𝑡 𝑟= log 𝑁 𝑛

Alice’s Encoder M 𝐼𝑓 𝐓=0, 𝐗 = 𝟎 𝐼𝑓 𝐓=1, 𝐗 =𝐸𝑛𝑐(𝐌) Bob’s Decoder 𝐗 𝐘 𝑏 BSC(pb) 𝐌 =𝐷𝑒𝑐( 𝐘 𝑏 ) 𝐌 T 1−𝜖 𝑟𝑒𝑙𝑖𝑎𝑏𝑙𝑒 Pr 𝐌 =𝐌 >1−𝜖 Message 𝐌∈{1, …, 𝑁} Trans. Status 𝐓∈{0, 1} 𝑁= 2 𝜃( 𝑛 )

Alice’s Encoder M 𝐼𝑓 𝐓=0, 𝐗 = 𝟎 𝐼𝑓 𝐓=1, 𝐗 =𝐸𝑛𝑐(𝐌) Bob’s Decoder 𝐗 𝐘 𝑏 BSC(pb) 𝐌 =𝐷𝑒𝑐( 𝐘 𝑏 ) 𝐌 T 1−𝜖 𝑟𝑒𝑙𝑖𝑎𝑏𝑙𝑒 Pr 𝐌 =𝐌 >1−𝜖 Message 𝐌∈{1, …, 𝑁} Trans. Status 𝐓∈{0, 1} 𝑁= 2 𝜃( 𝑛 ) BSC(pw) 𝐘 𝑤 𝐓 =𝐷𝑒𝑐( 𝐘 𝑤 ) Willie’s (Best) Estimator 𝐓  

Bash, Goeckel & Towsley [1] Shared secret 𝑂( 𝑛 log 𝑛 ) bits AWGN channels But capacity only 𝑂 𝑛 bits [1] B. A. Bash, D. Goeckel and D. Towsley, “Square root law for communication with low probability of detection on AWGN channels,” in Proceedings of the IEEE International Symposium on Information Theory (ISIT), 2012, pp. 448–452.

This work No shared secret BSC(pb) pb < pw BSC(pw)

Alice’s Encoder M 𝐼𝑓 𝐓=0, 𝐗 = 𝟎 𝐼𝑓 𝐓=1, 𝐗 =𝐸𝑛𝑐(𝐌) Bob’s Decoder 𝐗 𝐘 𝑏 BSC(pb) 𝐌 =𝐷𝑒𝑐( 𝐘 𝑏 ) 𝐌 T 1−𝜖 𝑟𝑒𝑙𝑖𝑎𝑏𝑙𝑒 Pr 𝐌 =𝐌 >1−𝜖 Message 𝐌∈{1, …, 𝑁} Trans. Status 𝐓∈{0, 1} 𝑁= 2 𝜃( 𝑛 ) BSC(pw) 𝐘 𝑤 𝐓 =𝐷𝑒𝑐( 𝐘 𝑤 ) Willie’s (Best) Estimator 𝐓  

Alice’s Transmission Status Hypothesis Testing Willie’s Estimate Alice’s Transmission Status 𝛼=Pr 𝐓 =1 𝐓=0 , 𝛽=Pr 𝐓 =0 𝐓=1

Alice’s Transmission Status Hypothesis Testing Willie’s Estimate Alice’s Transmission Status  

Alice’s Transmission Status Hypothesis Testing Willie’s Estimate Alice’s Transmission Status  

Alice’s Transmission Status Hypothesis Testing Willie’s Estimate Alice’s Transmission Status  

Intuition 𝐓=0, 𝐲 𝑤 = 𝐳 𝑤 ~Binomial(𝑛, 𝑝 𝑤 )

Intuition 𝐓=0, 𝐲 𝑤 = 𝐳 𝑤 ~Binomial 𝑛, 𝑝 𝑤 𝑊ℎ𝑒𝑛 𝐓=1,

Theorem 1 (Wt(c.w.)) (high deniability => low weight codewords) Too many codewords with weight “much” greater than 𝑐 𝑛 , then the system is “not very” deniable

Theorems 2 & 3 (Converse & achievability for reliable & deniable comm

Theorems 2 & 3 𝑝 𝑤 1/2 pb>pw 𝑝 𝑏 1/2

Theorems 2 & 3 𝑝 𝑤 1/2 𝑁=0 𝑝 𝑏 1/2

Theorems 2 & 3 𝑝 𝑤 pw=1/2 1/2 𝑝 𝑏 1/2

Theorems 2 & 3 𝑝 𝑤 1/2 (BSC(pb)) 𝑝 𝑏 1/2

Theorems 2 & 3 𝑝 𝑤 1/2 pb=0 𝑝 𝑏 1/2

Theorems 2 & 3 𝑝 𝑤 𝑁 = 2 𝑂( 𝑛 log 𝑛 ) , 𝑛 𝑛 = 2 𝑂( 𝑛 log 𝑛 ) 1/2 𝑝 𝑏 𝑝 𝑏 1/2

Theorems 2 & 3 𝑝 𝑤 1/2 pw>pb 𝑝 𝑏 1/2

Theorems 2 & 3 𝑝 𝑤 𝑁 = 2 𝑂( 𝑛 ) 1/2 “Standard” IT inequalities + 𝑁 = 2 𝑂( 𝑛 ) 1/2 “Standard” IT inequalities + Wt(“most codewords”)<√n (Thm 1) 𝑝 𝑏 1/2

Theorems 2 & 3 Main thm: 𝑝 𝑤 1/2 Achievable region 𝑁 = 2 Ω ( 𝑛 ) 𝑝 𝑏 𝑁 = 2 Ω ( 𝑛 ) 𝑝 𝑏 1/2

log 𝑛 𝑛/2 ≈𝑛 logarithm of # codewords n 𝑤 𝑡 𝐻 ( 𝒚 𝑤 )

log(# codewords) 𝑛𝐻( 𝑝 𝑤 ) 𝐱 = 0 Pr 𝐙 𝑤 ⁡(𝑤 𝑡 𝐻 𝐲 𝑤 ) 𝑂( 1 𝑛 ) 𝑝 𝑤 𝑛 𝑝 𝑤 𝑛+𝑂( 𝑛 ) n 𝑤 𝑡 𝐻 ( 𝐲 𝑤 ) 𝑝 𝑤 𝑛−𝑂( 𝑛 )

log(# codewords) 𝑛𝐻( 𝑝 𝑤 ∗𝜌) 𝑐 𝑛 Pr 𝐌, 𝐙 𝑤 ⁡(𝑤 𝑡 𝐻 𝐲 𝑤 ) 𝑂( 1 𝑛 ) n 𝑤 𝑡 𝐻 ( 𝐲 𝑤 ) (𝑝 𝑤 ∗𝜌)𝑛−𝑂( 𝑛 ) (𝑝 𝑤 ∗𝜌)𝑛 (𝑝 𝑤 ∗𝜌)𝑛+𝑂( 𝑛 )

Theorem 3 – Reliability proof sketch Weight 𝑂( 𝑛 ) Random code 1000001000000000100100000010000000100 0001000000100000010000000010000000001 0010000100000001010010000000100010011 . 2 𝑂( 𝑛 ) codewords 0000100000010000000000010000000010000

Theorem 3 – Reliability proof sketch Weight 𝑂( 𝑛 ) E(Intersection of 2 codewords) = O(1) “Most” codewords “well-isolated” 1000001000010000100100000010000000100 0001000000100000010000000010000000001 0010000100000001010010000000100010011 . 0000100000010000000000010000000010000

Theorem 3 – dmin decoding x + 𝑂( 𝑛 ) x’ Pr(x decoded to x’) < 2 −𝑂( 𝑛 )

Theorem 3 – Deniability proof sketch Recall: want to show 𝑉 𝐏 0 , 𝐏 1 <𝜖

log(# codewords) 𝑛𝐻( 𝑝 𝑤 ∗𝜌) 𝑐 𝑛 Pr 𝐌, 𝐙 𝑤 ⁡(𝑤 𝑡 𝐻 𝐲 𝑤 ) 𝑂( 1 𝑛 ) n 𝑤 𝑡 𝐻 ( 𝐲 𝑤 ) (𝑝 𝑤 ∗𝜌)𝑛−𝑂( 𝑛 ) (𝑝 𝑤 ∗𝜌)𝑛 (𝑝 𝑤 ∗𝜌)𝑛+𝑂( 𝑛 )

Theorem 3 – Deniability proof sketch Recall: want to show 𝑉 𝐏 0 , 𝐏 1 <𝜖 𝐏 0 𝐏 1

Theorem 3 – Deniability proof sketch log(# codewords) n Pr 𝑪, 𝐙 𝑤 ⁡(𝑤 𝑡 𝐻 𝐲 𝑤 ) 𝑂( 1 𝑛 ) 𝑤 𝑡 𝐻 ( 𝐲 𝑤 ) (𝑝 𝑤 ∗𝜌)𝑛−𝑂( 𝑛 ) (𝑝 𝑤 ∗𝜌)𝑛 (𝑝 𝑤 ∗𝜌)𝑛+𝑂( 𝑛 )

Theorem 3 – Deniability proof sketch logarithm of # codewords n 𝑤 𝑡 𝐻 ( 𝒚 𝑤 )

Theorem 3 – Deniability proof sketch 𝑬 𝑪 (𝐏 1 )!!! 𝐏 0 𝐏 1

Theorem 3 – Deniability proof sketch 𝑉 𝐏 0 , 𝐏 1 ≤𝑉 𝐏 0 , 𝑬 𝑪 (𝐏 1 ) +𝑉 𝑬 𝑪 (𝐏 1 ), 𝐏 1 𝑬 𝑪 (𝐏 1 )!!! 𝐏 0 𝐏 1

Theorem 3 – Deniability proof sketch 𝑬 𝑪 (𝐏 1 ) 𝐏 1

Theorem 3 – Deniability proof sketch logarithm of # codewords 𝑝 𝑤 𝑛 𝑝 𝑤 𝑛+𝑂( 𝑛 ) n 𝑤 𝑡 𝐻 ( 𝒚 𝑤 ) 𝑝 𝑤 𝑛−𝑂( 𝑛 )

Theorem 4 logarithm of # codewords n 𝑤 𝑡 𝐻 ( 𝒚 𝑤 )

Theorem 4 Too few codewords => Not deniable logarithm of n 𝑤 𝑡 𝐻 ( 𝒚 𝑤 )

Summary

Summary