Some Goodness Properties of LDA Lattices

Transcription

1 Some Goodness Properties of LDA Lattices Shashank Vatedka and Navin Kashyap Department of ECE Indian Institute of Science Bangalore, India Information Theory Workshop Jerusalem, Israel 25 / 28

2 Lattices for communication over Gaussian channels Some problems where lattices yield optimal/near optimal solutions: Sphere packing/covering Quantization AWGN channel Dirty paper channel Symmetric Gaussian interference channel Bidirectional relaying/physical layer network coding Physical-layer security and more 2 / 28

3 Lattices in R n V(Λ) Fundamental Voronoi region: V(Λ) 3 / 28

4 Lattices in R n V(Λ) r pack (Λ) Fundamental Voronoi region: V(Λ) Packing radius: r pack (Λ) 3 / 28

5 Lattices in R n V(Λ) r eff (Λ) Fundamental Voronoi region: V(Λ) Packing radius: r pack (Λ) Effective radius: r eff (Λ) 3 / 28

6 Lattices in R n V(Λ) r cov (Λ) Fundamental Voronoi region: V(Λ) Packing radius: r pack (Λ) Effective radius: r eff (Λ) Covering radius: r cov (Λ) Clearly, r pack (Λ) r eff (Λ) r cov (Λ) 3 / 28

7 Lattices in R n r pack (Λ) r cov (Λ) V(Λ) r eff (Λ) Fundamental Voronoi region: V(Λ) Packing radius: r pack (Λ) Effective radius: r eff (Λ) Covering radius: r cov (Λ) Clearly, r pack (Λ) r eff (Λ) r cov (Λ) Normalized second moment (NSM)/Normalized moment of inertia (NMI): G(Λ) Let X Unif(V(Λ)) Then, G(Λ) = vol(v(λ)) 2/n n E X 2 3 / 28

8 Components of a lattice code Lattice Λ (In general, could be a coset) Shaping region S Convex set Lattice code: C = Λ S 4 / 28

9 Nested lattice codes n-dimensional lattices Λ, Λ such that Λ Λ Λ : coarse lattice Λ: fine lattice 5 / 28

10 Nested lattice codes n-dimensional lattices Λ, Λ such that Λ Λ Λ : coarse lattice Λ: fine lattice Shaping region: V(Λ ) R = n log Λ V(Λ ) = n log vol(v(λ )) vol(v(λ)) 5 / 28

11 Nested lattice codes n-dimensional lattices Λ, Λ such that Λ Λ Λ : coarse lattice Λ: fine lattice Shaping region: V(Λ ) R = n log Λ V(Λ ) = n log vol(v(λ )) vol(v(λ)) With dithered transmission: P = n E X 2 = G(Λ )(vol(v(λ ))) 2/n With CLP decoding: P e = Pr[Z noise / V(Λ)] 5 / 28

12 Properties that we are looking for {Λ (n) } is good for V(Λ) r pack (Λ) r eff (Λ) r cov (Λ) packing if lim n r pack (Λ (n) ) r eff (Λ (n) ) 2 6 / 28

13 Properties that we are looking for {Λ (n) } is good for r pack (Λ) r eff (Λ) r cov (Λ) V(Λ) packing if lim n r pack (Λ (n) ) r eff (Λ (n) ) 2 MSE (mean squared error) quantization if lim n G(Λ (n) ) = 2πe 6 / 28

14 Properties that we are looking for {Λ (n) } is good for r pack (Λ) r eff (Λ) r cov (Λ) V(Λ) packing if lim n r pack (Λ (n) ) r eff (Λ (n) ) 2 MSE (mean squared error) quantization if lim n G(Λ (n) ) = 2πe covering if lim n r cov (Λ (n) ) r eff (Λ (n) ) = 6 / 28

15 Properties that we are looking for {Λ (n) } is good for r pack (Λ) r eff (Λ) r cov (Λ) V(Λ) packing if lim n r pack (Λ (n) ) r eff (Λ (n) ) 2 MSE (mean squared error) quantization if lim n G(Λ (n) ) = 2πe covering if lim n r cov (Λ (n) ) r eff (Λ (n) ) = channel coding if Pr[Z (n) / V(Λ (n) )] as n for every semi norm-ergodic noise (eg, AWGN) vectors Z (n) with σ 2 := n E Z(n) that satisfy vol(λ(n) ) 2/n > 2πeσ 2 6 / 28

16 Nested lattices for communication over Gaussian channels Suppose we can construct nested lattice pairs such that: {Λ (n) } good for channel coding {Λ (n) } good for MSE quantization/covering 7 / 28

17 Nested lattices for communication over Gaussian channels Suppose we can construct nested lattice pairs such that: {Λ (n) } good for channel coding {Λ (n) } good for MSE quantization/covering This can be used to construct codes that achieve the capacity of the AWGN channel and the dirty paper channel (Erez and Zamir 24; Ordentlich and Erez 22) achieve rates within a constant gap of the capacity of the bidirectional relay (Wilson et al 2; Nazer and Gastpar 2a; Nam et al 2; Ordentlich and Erez 22) achieve rates guaranteed by many lattice-based physical-layer network coding schemes for Gaussian networks (Nazer and Gastpar 2b; Zhang et al 26) Also important components in coding schemes for secure communication over the Gaussian wiretap channel (Ling et al 24) and the bidirectional relay (He and Yener 23; Vatedka et al 24) 7 / 28

18 Construction A Let C be an (n, k) linear code over F p, for prime p The Construction-A lattice Λ A (C) is defined as Λ A (C) = {x Z n : x c mod p for some c C} 8 / 28

19 Construction A Let C be an (n, k) linear code over F p, for prime p The Construction-A lattice Λ A (C) is defined as Λ A (C) = {x Z n : x c mod p for some c C} Considering {,,, p } to a subset of Z, we may view C as a subset of Z n Then, Λ A (C) = C + pz n Example: C is a (2, ) linear code over F 3 with generator matrix [ 2] 8 / 28

20 Construction A Let C be an (n, k) linear code over F p, for prime p The Construction-A lattice Λ A (C) is defined as Λ A (C) = {x Z n : x c mod p for some c C} Considering {,,, p } to a subset of Z, we may view C as a subset of Z n Then, Λ A (C) = C + pz n Example: C is a (2, ) linear code over F 3 with generator matrix [ 2] 8 / 28

21 Construction A Let C be an (n, k) linear code over F p, for prime p The Construction-A lattice Λ A (C) is defined as Λ A (C) = {x Z n : x c mod p for some c C} Considering {,,, p } to a subset of Z, we may view C as a subset of Z n Then, Λ A (C) = C + pz n Example: C is a (2, ) linear code over F 3 with generator matrix [ 2] 8 / 28

22 Goodness of Construction-A lattices The (n, k, p) ensemble: For prime p, the ensemble of all Construction-A lattices obtained from (n, k) linear codes over F p Theorem (Erez et al 25, Thm 5) Let k and p be suitably chosen functions of n If Λ (n) is a randomly chosen Construction-A lattice from an (n, k, p) ensemble, then, for every ϵ >, r pack (Λ (n) ) r eff (Λ (n) ) 2 ϵ, r cov (Λ (n) ) r eff (Λ (n) ) + ϵ, G(Λ (n) ) 2πe + ϵ, Pr[Z(n) / V(Λ (n) )] ϵ with probability tending to as n 9 / 28

23 Goodness of Construction-A lattices The (n, k, p) ensemble: For prime p, the ensemble of all Construction-A lattices obtained from (n, k) linear codes over F p Theorem (Erez et al 25, Thm 5) Let k and p be suitably chosen functions of n If Λ (n) is a randomly chosen Construction-A lattice from an (n, k, p) ensemble, then, for every ϵ >, r pack (Λ (n) ) r eff (Λ (n) ) 2 ϵ, r cov (Λ (n) ) r eff (Λ (n) ) + ϵ, G(Λ (n) ) 2πe + ϵ, Pr[Z(n) / V(Λ (n) )] ϵ with probability tending to as n How do we decode?? 9 / 28

24 Lattices with Low Decoding Complexity Some constructions of good lattices with low-complexity decoding algorithms Low Density Construction-A (LDA) Lattices: Construction A on nonbinary LDPC codes (di Pietro et al 22) LDPC lattices: Construction D on nested binary LDPC codes (Sadeghi et al 26) Turbo lattices: Construction D on turbo codes (Sakzad et al 2) Polar lattices: Construction D on nested polar codes (Yan et al 23) Low-Density Lattice Codes (LDLC): Not obtained from linear codes; The dual lattice has a low-density generator matrix (Sommer et al 28) / 28

25 Low-density Construction-A (LDA) lattices Use ( V, C )-regular LDPC codes to obtain Construction-A lattices We can then use BP decoding! L V C R (di Pietro et al 22): introduced logarithmic degree LDA lattices, showed goodness for channel coding (di Pietro et al 23a): constant degree LDA lattices are good for channel coding (di Pietro et al 23b) and (Tunali et al 23): simulations showing that LDA lattices go close to Poltyrev limit with BP decoding n left vertices n( R) right vertices (di Pietro 24): nested LDA lattices achieve capacity of AWGN channel with CLP decoding / 28

26 The Tanner graph To obtain good lattices, we want the Tanner graph of the underlying LDPC code to satisfy certain expansion properties: There exist positive constants ϵ, ϑ, and α A and β B such that V C L R n left vertices n( R) right vertices 2 / 28

27 The Tanner graph To obtain good lattices, we want the Tanner graph of the underlying LDPC code to satisfy certain expansion properties: There exist positive constants ϵ, ϑ, and α A and β B such that L R Left vertex expansion: for any S L S ϵn = N(S) A S S = N(S) α S n( R) 2α S N(S) 2 / 28

28 The Tanner graph To obtain good lattices, we want the Tanner graph of the underlying LDPC code to satisfy certain expansion properties: There exist positive constants ϵ, ϑ, and α A and β B such that L R Left vertex expansion: for any S L S ϵn = N(S) A S S = N(S) α S n( R) 2α T Right vertex expansion: for any T R T ϑn( R) = N(T ) B T T n( R) 2 = N(T ) β T N(T ) 2 / 28

29 Random graphs are good expanders Fix constants α A and R < β min ( 2 R, B ), and < ϑ, ϵ < /2 Lemma (di Pietro 24, Lem 33) If V > f (α, A, β, B, ϵ, ϑ, R), then the probability that a ( V, V ( R))-regular graph is a good expander goes to one as n Throughout, we assume that the hypothesis is satisfied, and the Tanner graph is a good expander 3 / 28

30 The (G, λ) ensemble of LDA lattices Let λ > and p be the smallest prime greater than n λ Pick a ( V, V ( R))-regular G which is a good expander Let Ĥ be the corresponding adjacency (parity-check) matrix Ĥ =, Replace the s with iid Unif(F p ) rvs h h 2 h 5 h H = 22 h 24 h 26 h 33 h 34 h 35 h 4 h 43 h 46 Apply Construction A on the code defined by H 4 / 28

31 Nested LDA lattices H n( R ) n H first n( R) rows define C n( R ) rows define C Pick nested LDPC codes C and C, with C C Then, Λ = Λ A (C) and Λ = Λ A (C ) 5 / 28

32 Parameters of the LDA ensemble We choose the parameters so as to satisfy: < R < < α A A > 2( + R) ϵ = R A+ R { } R < β min 2 B, R B > 2 (+R) ( R) ϑ = B( R)+ V > f (α, A, β, B, ϵ, ϑ, R) For example, R = /3 α = 27 A = 3 ϵ = 82 β = 6 B = 5 ϑ = 23 V = 2 6 / 28

33 Goodness for channel coding Recall: p n λ The following result was proved by di Pietro: Theorem (di Pietro 24, Theorem 32) Let Λ be a lattice chosen uniformly at random from a (G, λ) LDA ensemble If { } λ > max 2(α + R), 3 2(A + R),, B( R) then the probability that Λ is good for channel coding tends to as n 7 / 28

34 Goodness for channel coding Recall: p n λ The following result was proved by di Pietro: Theorem (di Pietro 24, Theorem 32) Let Λ be a lattice chosen uniformly at random from a (G, λ) LDA ensemble If { } λ > max 2679, 6429, 4286, then the probability that Λ is good for channel coding tends to as n 7 / 28

35 Goodness for channel coding Recall: p n λ The following result was proved by di Pietro: Theorem (di Pietro 24, Theorem 32) Let Λ be a lattice chosen uniformly at random from a (G, λ) LDA ensemble If { } λ > max 2679, 6429, 4286, then the probability that Λ is good for channel coding tends to as n Also, (di Pietro 24) found sufficient conditions on parameters for nested LDA lattices to achieve capacity of AWGN channel 7 / 28

36 Packing Goodness Proofs of channel coding goodness and packing goodness are very similar Theorem (Vatedka & Kashyap, ITW 5) Let Λ be a lattice chosen uniformly at random from a (G, λ) LDA ensemble, Furthermore, let a { λ > max 2(α + R), 3 2(A + R), B( R) Then, the probability that Λ is good for packing tends to as n } a p n λ 8 / 28

37 Packing Goodness Proofs of channel coding goodness and packing goodness are very similar Theorem (Vatedka & Kashyap, ITW 5) Let Λ be a lattice chosen uniformly at random from a (G, λ) LDA ensemble, Furthermore, let a λ > max { 2679, 6429, 4286 } Then, the probability that Λ is good for packing tends to as n a p n λ 8 / 28

38 Goodness for MSE quantization Theorem (Vatedka & Kashyap, ITW 5) Suppose a λ > max { R, R, 2 A 2( + R), 2 B( R) 2( + R), ( 2 AB ) } A Let Λ be randomly chosen from a (G, λ) LDA ensemble Then, the probability that Λ is good for MSE quantization tends to as n a p n λ 9 / 28

39 Goodness for MSE quantization Theorem (Vatedka & Kashyap, ITW 5) Suppose a λ > max { 3, 5, 6, 3, 336 } Let Λ be randomly chosen from a (G, λ) LDA ensemble Then, the probability that Λ is good for MSE quantization tends to as n a p n λ 9 / 28

40 Packing goodness of the duals Motivation: Perfect secrecy in an honest-but-curious bidirectional relay setting To achieve best known rates in presence of Gaussian noise, need (Vatedka et al 24; Vatedka and Kashyap 25) {Λ (n) } to be good for channel coding {Λ (n) } to be good for MSE quantization duals of {Λ (n) } to be good for packing Theorem (Vatedka & Kashyap, ITW 5) If { λ > max 2( R), 2B + 3/2 B( R) then the dual of a randomly chosen lattice from a (G, λ) LDA ensemble is good for packing with probability tending to as n }, 2 / 28

41 Packing goodness of the duals Motivation: Perfect secrecy in an honest-but-curious bidirectional relay setting To achieve best known rates in presence of Gaussian noise, need (Vatedka et al 24; Vatedka and Kashyap 25) {Λ (n) } to be good for channel coding {Λ (n) } to be good for MSE quantization duals of {Λ (n) } to be good for packing Theorem (Vatedka & Kashyap, ITW 5) If { } λ > max 75, 4928, then the dual of a randomly chosen lattice from a (G, λ) LDA ensemble is good for packing with probability tending to as n 2 / 28

42 Complexity So far, assumed CLP decoder was used Using BP decoder, complexity would be O(np log p) We want p to be as small as possible Turns out we cannot make p any less than n 4 for our proofs to go through Although this means order of complexity is polynomial in n, still too high! Simulation results in (di Pietro et al 23b; Tunali et al 23) used much smaller values of p (p = for blocklength ) but obtained good performance Better proof techniques required 2 / 28

43 LDA lattices: an attractive option Has a natural low-complexity decoding algorithm! Good for packing Good for MSE quantization Good for channel coding Duals are good for packing Open: Are LDA lattices good for covering? Error exponents with CLP decoder? Performance with BP decoding? Full version: 22 / 28

44 References I U Erez and R Zamir, Achieving /2 log (+ SNR) on the AWGN channel with lattice encoding and decoding, IEEE Transactions on Information Theory, vol 5, no, pp , 24 O Ordentlich and U Erez, A simple proof for the existence of good pairs of nested lattices, in 22 IEEE 27th Convention of Electrical & Electronics Engineers in Israel (IEEEI), IEEE, 22, pp 2 M Wilson, K Narayanan, H Pfister, and A Sprintson, Joint physical layer coding and network coding for bidirectional relaying, IEEE Transactions on Information Theory, vol 56, no, pp , 2 23 / 28

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46 References III C Ling, L Luzzi, J-C Belfiore, and D Stehle, Semantically secure lattice codes for the Gaussian wiretap channel, IEEE Transactions on Information Theory, vol 6, no, pp , 24 X He and A Yener, Strong secrecy and reliable byzantine detection in the presence of an untrusted relay, IEEE Transactions on Information Theory, vol 59, no, pp 77 92, 23 S Vatedka, N Kashyap, and A Thangaraj, Secure compute-and-forward in a bidirectional relay, accepted, IEEE Transactions on Information Theory, 24 [Online] Available: U Erez, S Litsyn, and R Zamir, Lattices which are good for (almost) everything, IEEE Transactions on Information Theory, vol 5, no, pp , / 28

47 References IV N di Pietro, J Boutros, G Zemor, and L Brunel, Integer low-density lattices based on Construction A, in 22 IEEE Information Theory Workshop (ITW), 22, pp M-R Sadeghi, A Banihashemi, and D Panario, Low-density parity-check lattices: construction and decoding analysis, IEEE Transactions on Information Theory, vol 52, no, pp , 26 A Sakzad, M-R Sadeghi, and D Panario, Construction of turbo lattices, in 2 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton), 2, pp 4 2 Y Yan, C Ling, and X Wu, Polar lattices: where Arikan meets Forney, in 23 IEEE International Symposium on Information Theory (ISIT), 23, pp / 28

48 References V N Sommer, M Feder, and O Shalvi, Low-density lattice codes, IEEE Transactions on Information Theory, vol 54, no 4, pp , 28 N di Pietro, G Zemor, and J Boutros, New results on Construction-A lattices based on very sparse parity-check matrices, in 23 IEEE International Symposium on Information Theory (ISIT), 23, pp N di Pietro, J Boutros, G Zemor, and L Brunei, New results on low-density integer lattices, in 23 Information Theory and Applications Workshop (ITA), 23, pp 6 N Tunali, K Narayanan, and H Pfister, Spatially-coupled low density lattices based on Construction A with applications to compute-and-forward, in Information Theory Workshop (ITW), 23 IEEE, 23, pp 5 27 / 28

49 References VI N di Pietro, On infinite and finite lattice constellations for the additive white Gaussian noise channel, PhD thesis, University of Bordeaux, 24 S Vatedka and N Kashyap, Nested lattice codes for secure bidirectional relaying with asymmetric channel gains, in IEEE Information Theory Workshop, Jerusalem, Israel, / 28