Interactive Communication for Data Exchange

Transcription

1 Interactive Communication for Data Exchange Himanshu Tyagi Indian Institute of Science, Bangalore Joint work with Pramod Viswanath and Shun Watanabe

2 The Data Exchange Problem [ElGamal-Orlitsky 84], [Csiszár-Narayan 04] X Y by b X A protocol π constitutes an ɛ-data exchange (ɛ-de) protocol if ( ) Pr ˆX = X, Ŷ = Y 1 ɛ. What is the minimum length L ɛ (X, Y ) of an ɛ-de protocol? 1

3 The Slepian-Wolf Problem Only X needs to be sent to an observer of Y. [Slepian-Wolf 73] Optimal rate for the case of IID observations: Rɛ = H(X Y ), 0 < ɛ < 1. [Miyake-Kanaya 95] Single-shot bounds: L ɛ (X Y ) λ + log [1 ɛ Pr (h(x Y ) λ)] : lower bound L ɛ (X Y ) λ log [ɛ Pr (h(x Y ) λ)] : upper bound 2

4 The Slepian-Wolf Problem Only X needs to be sent to an observer of Y. [Slepian-Wolf 73] Optimal rate for the case of IID observations: Rɛ = H(X Y ), 0 < ɛ < 1. [Miyake-Kanaya 95] Single-shot bounds: L ɛ (X Y ) λ + log [1 ɛ Pr (h(x Y ) λ)] : lower bound L ɛ (X Y ) λ log [ɛ Pr (h(x Y ) λ)] : upper bound Prob. < 1 Prob. > Spectrum of h(x Y )= log P X Y (X Y ) 2

5 Can Interaction Help? Asymptotically, interaction does not help in the SW problem. 3

6 Can Interaction Help? Asymptotically, interaction does not help in the SW problem. Instances where interaction is known to help: [Orlitsky 90] Single-shot, worst-case length: One round of interaction is almost optimal without error [Feder-Shulman 02] Universal version, adaptive rate: An interactive protocol accomplishes this task [Yang-He 10] Single-shot, average length An interactive protocol attains roughly H(X Y ) 3

7 Can Interaction Help? Asymptotically, interaction does not help in the SW problem. Instances where interaction is known to help: [Orlitsky 90] Single-shot, worst-case length: One round of interaction is almost optimal without error [Feder-Shulman 02] Universal version, adaptive rate: An interactive protocol accomplishes this task [Yang-He 10] Single-shot, average length An interactive protocol attains roughly H(X Y ) Can interaction help in the data exchange problem? 3

8 Using Slepian-Wolf scheme for Data Exchange The following rate is achievable for the IID case: H(X Y ) def = H(X Y ) + H(Y X). 4

9 Using Slepian-Wolf scheme for Data Exchange The following rate is achievable for the IID case: [Csiszár-Narayan 04] H(X Y ) def = H(X Y ) + H(Y X). This rate is the least possible. The proof relies on a property of interactive communication: H(Π) H(Π X, U) + H(Π Y, V ). Implication: Noninteractive communication can attain the optimal rate 4

10 Using Slepian-Wolf scheme for Data Exchange The following rate is achievable for the IID case: [Csiszár-Narayan 04] H(X Y ) def = H(X Y ) + H(Y X). This rate is the least possible. The proof relies on a property of interactive communication: H(Π) H(Π X, U) + H(Π Y, V ). Implication: Noninteractive communication can attain the optimal rate Is interaction of any use for data exchange? 4

11 Main Result: Bounds on L ɛ (X, Y ) We show that interaction is indeed helpful. We characterize the min. length of interactive communication needed, thereby characterizing the gain due to interaction. 5

12 Main Result: Bounds on L ɛ (X, Y ) We show that interaction is indeed helpful. We characterize the min. length of interactive communication needed, thereby characterizing the gain due to interaction. Define sum conditional entropy h(x Y ) def = h(x Y ) + h(y X) Theorem (Single-shot) For every 0 < ɛ < 1, we have L ɛ (X, Y ) λ log [ɛ Pr (h(x Y ) λ)], λ > 0, L ɛ (X, Y ) λ + log [1 ɛ Pr (h(x Y ) λ)], λ > 0. 5

13 Corollary 1: Second-Order Asymptotics for IID Sources Let (X n, Y n ) = (X i, Y i ) n i=1 be IID realizations of (X, Y ). Theorem For every 0 < ɛ < 1, we have L ɛ (X n, Y n ) = nh(x Y ) + nvar[h(x Y )]Q 1 (ɛ) + o( n). 6

14 Corollary 1: Second-Order Asymptotics for IID Sources Let (X n, Y n ) = (X i, Y i ) n i=1 be IID realizations of (X, Y ). Theorem For every 0 < ɛ < 1, we have L ɛ (X n, Y n ) = nh(x Y ) + nvar[h(x Y )]Q 1 (ɛ) + o( n). This length is strictly smaller than that attained by noninteractive protocols. 6

15 Corollary 2: Minimum Rate for General Sources Let (X, Y) = (X n, Y n ) n=1 be a general source sequence. Define the minimum rate of communication for data exchange as R (X, Y) def 1 = inf lim sup {ɛ n} n n L ɛ n (X n, Y n ), where the infimum is over all sequences ɛ n 0. Theorem For a general source sequence (X, Y), R (X, Y) = H(X Y), where H(X Y) denotes the lim sup in probability of h(x n Y n ). 7

16 Corollary 2: Minimum Rate for General Sources Let (X, Y) = (X n, Y n ) n=1 be a general source sequence. Define the minimum rate of communication for data exchange as R (X, Y) def 1 = inf lim sup {ɛ n} n n L ɛ n (X n, Y n ), where the infimum is over all sequences ɛ n 0. Theorem For a general source sequence (X, Y), R (X, Y) = H(X Y), where H(X Y) denotes the lim sup in probability of h(x n Y n ). This rate is strictly smaller than that attained by noninteractive protocols. 7

17 Proof Sketch for the Converse

18 Digression: Secret Key Agreement X Y K x K y K constitutes an ɛ-secret key of length log K if where 1 2 P K xk yf P (2) unif P F 1 ɛ, P (2) unif(k x, k y ) = 1 K 1(k x = k y ). The maximum length of an ɛ-sk is denoted by S ɛ (X, Y ). 9

19 Main Heuristic Parties with correlated observations share more bits than what they communicate. The extra bits shared can be extracted as a secret key. Thus, if the parties share R shared bits and communicate R bits, R shared R S(X, Y ) R shared S(X, Y ) R [Csisár-Narayan 04] First formalized this duality to obtain SK capacity [T-Narayan-Gupta 10, T 12] characterization of secure computability 10

20 Warm-up: Optimal Rate for Data Exchange Csiszár-Narayan approach flipped around: Consider a rate R protocol for data exchnage. Both parties share roughly nh(xy ) bits at the end. Using an extractor lemma we can generate a SK of rate H(XY ) R, which must be less than the SK capacity I(X Y ). Thus, R H(XY ) I(X Y ) = H(X Y ) + H(Y X). 11

21 Warm-up: Optimal Rate for Data Exchange Csiszár-Narayan approach flipped around: Consider a rate R protocol for data exchnage. Both parties share roughly nh(xy ) bits at the end. Using an extractor lemma we can generate a SK of rate H(XY ) R, which must be less than the SK capacity I(X Y ). Thus, R H(XY ) I(X Y ) = H(X Y ) + H(Y X). We seek to extend this argument to a single-shot setup. 11

22 Upper Bound for Secret Key Length [T-Watanabe 14] Theorem For every 0 < ɛ, η < 1 with η < 1 ɛ, we have S ɛ (X, Y ) λ log (P λ ɛ η) + 2 log 1/η, λ > 0, where P λ = P XY ({(x, y) : log }) P XY (x, y) Q X (x) Q Y (y) < λ. 12

23 Converse for Almost Uniform Sources Consider a data exchange protocol of length l. H min (XY ) H max (XY ) Spectrum of h(xy ) Using the Leftover Hash Lemma, we can extract a SK of length H min (XY ) l. 13

24 Converse for Almost Uniform Sources Consider a data exchange protocol of length l. H min (XY ) H max (XY ) Spectrum of h(xy ) Using the upper bound for S ɛ (X, Y ), H min (XY ) l λ log (P XY (i(x Y ) < λ) ɛ) = λ log (P XY (h(xy ) h(x Y ) < λ) ɛ) H max (XY ) γ log (P XY (h(x Y ) > γ) ɛ) 13

25 Converse for Almost Uniform Sources Consider a data exchange protocol of length l. H min (XY ) H max (XY ) Spectrum of h(xy ) Thus, l H max (XY ) H min (XY ) + γ + log (P XY (h(x Y ) > γ) ɛ), which gives the converse bound if H max (XY ) H min (XY ). 13

26 General Converse via Spectrum Slicing Slice the spectrum into N slices of width each. H min (XY ) H max(xy ) E j Spectrum of h(xy ) There exists a slice E j with P XY (E j ) N 2, and so P XY P XY Ej N 2 P XY. The proof is completed by applying the previous bound to P XY Ej. 14

27 Our Achievability Scheme

28 Rough Sketch of Our Scheme H min (X Y ) h(x Y ) T j h i random binning of X into H ξ min (X Y ) values, i = 1, random binning of X into values, 2 i N. First party sends bin indices Π i = h i (x) successively until it receives an ACK or i = N Second party sends an ACK when it finds an ˆx s.t. (ˆx, y) T i and h j (ˆx) = Π j, 1 j i. 16

29 In Closing... Prob. < Spectrum of h(x4y ) The minimum length of communication for ɛ-data exchange is equal to roughly the ɛ-tail λ ɛ of h(x Y ). Interaction is necessary to attain this rate. 17