Construction πA Lattices: AReviewandRecent Results · 2019. 12. 20. · • Secrecy - He & Yener,...

Construction πA Lattices: A Review and Recent

Results

Yu-Chih (Jerry) Huang

Department of Communication Engineering

National Taipei University

Joint work with Krishna Narayanan @ Texas A&M University

1 / 38

Lattice codes are everywhere

(Nested) Lattice codes have many applications in IT

• See paper “Lattices are Everywhere” by Zamir

• Single user Gaussian channel - Erez & Zamir

• Coding with side information (Wyner-Ziv and Costa) - Zamir, Erez & Shamai

• Physical layer network coding - Wilson et al, Nam et al

• Secrecy - He & Yener, Ling et al

• Interference alignment - Sridharan, Jafaraian, Vishwanath & Jafar andOrdentlich, Erez, & Nazer

• Dirty multiple access channel - Philosof, Khisti, Erez & Zamir

• Compute-and-forward - Nazer & Gastpar

Most of these results are based on Construction A lattices

2 / 38












Generate asymptotically good lattices; but comes with large decoding complexity

2 / 38












Generate asymptotically good lattices; but comes with large decoding complexity

Main theme: Construction that generates good lattices with low complexity

2 / 38

My view of these constructions

After Alister’s talk on Monday, I was like

3 / 38

My view of these constructions

Compute-and-forward

Nazer-Gastpar 09

Algebraic approach

Feng-Silva-Kschichang 11

Multistage CFHuang-Narayanan 14

Elementary divisorWang-Burr 15

Structure theorem forfinitely generated modules

Structure theorem forfinitely generated modules

Chinese

remainder

theorem

Chinese

remainder

theorem

(rings) (modules)

Construction πA is special case of EDC where we can show interesting things

3 / 38

Lattices

n-dimensional lattice Λn: A discrete subgroup of Rn

• Can be expressed by generator matrix G as

Λn = {Gz : z ∈ Zn}

• Closed underAddition: λ1, λ2 ∈ Λn implies λ1 + λ2 ∈ Λn

Reflection: λ1 ∈ Λn implies −λ1 ∈ Λn

−5 0 5−5

−4

−3

−2

−1

0

1

2

3

4

5

Figure: Rectangular (Z2) lattice, Gaussianintegers Z[i]

−5 0 5−5

−4

−3

−2

−1

0

1

2

3

4

5

Figure: Hexagonal (A2) lattice, Eisensteinintegers Z[ω] 4 / 38

Lattices

• Lattice quantizer: For x ∈ Rn, QΛ(x) , argmin

λ∈Λ‖x− λ‖2

• Fundamental Voronoi region: V(Λ) , {x : QΛ(x) = 0}• Modulo operation: For x ∈ R

n, x mod Λ , x−QΛ(x)

−5 0 5−5

−4

−3

−2

−1

0

1

2

3

4

5

5 / 38

Goodness for MSE quantization

• Let U ∼Uniform(V)

• Second moment per dim associated with Λ

σ2(Λ) ,1

nE‖U‖2 =

1

n

∫

V‖x‖2dx

Vol(Λ)

• Normalized second moment (NSM) of Λ

G(Λ) ,σ2(Λ)

Vol(Λ)2

n

>1

2πe

• Note that rB has G(rB) → 1/(2πe)

• A seq of Λn is good for MSE quantization if has G(Λ) → 1/(2πe)

• Related to performance of lattice quantizer at high resolution

6 / 38

Goodness for channel coding

rpack

V(Λ)

rz =√

nσ2z

reff

• Consider using Λ as our transmitted constellation, no power constraint, overAWGN channel

y = λ+ z, z ∼ N(0, σ2zI)

• Use lattice decoding, i.e., decoding to QΛ(y)

• From LLN, z lies inside rzB w.h.p.

• A seq of lattices is good for channel coding if pe → 0 whenever reff > rz

7 / 38

Outline

1 Construction A Lattices Review

2 Construction πA Multilevel Lattices

3 Application 1: Multistage Compute-and-Forward

4 Application 2: Lattice Index Coding

5 Future Directions

8 / 38

Construction A Lattices Review

Outline





5 Future Directions

9 / 38


N -Dim Lattice Using Construction A (Leech-Sloane 71)

• C linear code over Fp

• M : Fp → Z natural mapping

• Λ = M(C) + pZN

C Mw ∈ F

mp

+

v ∈ pZN

λ ∈ Λ

x ∈ (Z/pZ)Nc ∈ FNp

0 2 4 6 8 10 12

0

2

4

6

8

10

12

10 / 38


N -Dim Lattice Using Construction A (Leech-Sloane 71)

• C linear code over Fp

• M : Fp → Z natural mapping

• Λ = M(C) + pZN

• λ ∈ Λ iff λ mod pZN ∈ C C Mw ∈ F

mp

+

v ∈ pZN

λ ∈ Λ

x ∈ (Z/pZ)Nc ∈ FNp

0 2 4 6 8 10 12

0

2

4

6

8

10

12

0 5 10 15 20 250

5

10

15

20

25

10 / 38


Construction A: What makes this a Lattice?

Construction A: Λ = M(C) + pZN

• λ1 = M(c1) + pk1 and λ2 = M(c2) + pk2

• λ1 + λ2 = M(c1) +M(c2) + pk1 + pk2

• It becomes M(c1 ⊕ c2) + pk3

• Thus, (λ1 + λ2) mod pZN = M(c1 ⊕ c2)

0 5 10 15 20 250

5

10

15

20

25

11 / 38


Construction A: What makes this a Lattice?

Construction A: Λ = M(C) + pZN

• λ1 = M(c1) + pk1 and λ2 = M(c2) + pk2

• λ1 + λ2 = M(c1) +M(c2) + pk1 + pk2

• It becomes M(c1 ⊕ c2) + pk3

• Thus, (λ1 + λ2) mod pZN = M(c1 ⊕ c2)

0 5 10 15 20 250

5

10

15

20

25

Natural mapping and mod p preserve ring structures between Z and Fp

11 / 38


Why Construction A with natural mapping would work?

Λ = M(C) + pZN

• M−1((λ1 + λ2) mod pZ) = c1 ⊕ c2

• This would work if M−1 ◦ mod pZ is ring homomorphism

12 / 38


Why Construction A with natural mapping would work?

Λ = M(C) + pZN

• M−1((λ1 + λ2) mod pZ) = c1 ⊕ c2

• This would work if M−1 ◦ mod pZ is ring homomorphism

Quotient ring

• pZ is an ideal in Z

• Coset decomposition Z/pZ results in a quotient ring

• For prime p, pZ is maximal ideal (since Z is PID)

• Z/pZ ∼= Fp

1 201211 4 531098

0 1 2 3-4 -3 -2 -1 4 5-5 6 7 8 9 10-9 -8 -7 -6-10

6 7 9 108

11 12 13

12 0111 20 4 53 6 7

-12-11-13

Z/13Z

12 / 38


Generalizations

Construction A over Z: M natural mapping

• Natural mapping happens to be ring isomorphism

• mod p is canonical ring homomorphism

• ϕ , M−1 ◦ mod p is ring homo

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Generalizations





• This is all the math I am going to use

Generalizations:

13 / 38


Generalizations





• This is all the math I am going to use

Generalizations:

• Construction A over Z[ω]Increase average computation rates for compute-and-forward (CF)Construction and application to CF: T-IT 15

• Go beyond PID, Construction A over OK

Consider only rings of imaginary quadratic integersPropose adaptive CF where TX adaptively work with best OK

ISIT 15 and will be submitted to T-IT soon

• Construction πA

Can be used to decrease decoding complexityNaturally suited for broadcasting with receiver side informationITW 13, ISIT 14, T-IT submitted 15, under revision 16

13 / 38

Construction πA Multilevel Lattices

Outline





5 Future Directions

14 / 38


Motivation: Problem with Construction A

λ = M(c) + p · k where c ∈ C and k ∈ ZN

0 5 10 15 20 250

5

10

15

20

25

• k is unprotected; p has to be large to have a good lattice

• Complexity depends on decoding the linear code over Fp

• E.g., simulation results by di Pietro, Boutros, Zemor, Brunel with F11 and F41

15 / 38


Main result in this part

Construction πA based on the Chinese Remainder Theorem

• p does not have to be prime - can be replaced by p1p2 . . . pL

• Instead of working over Fp, we can work over Fp1× Fp2

× . . .× Fpl

• Ex: q = 210 with just codes over F2,F3,F5,F7

• Show existence of sequence of lattices that are optimal

In short: New construction of lattices that preserve algebraic structures andgoodness with substantially lower complexity

16 / 38


Chinese Remainder Theorem for Commutative Rings

Let p1, . . . , pL be distinct primes, q = p1 · p2 . . . pL, and ql = q/pl

Z/qZ ∼= Zp1× Zp2

× . . .× ZpL∼= Fp1

× Fp2× . . .× FpL

An isomorphism:

M(v1, . . . , vL) = s1q1v1 + . . .+ sLqLv

L mod qZ,

where s1, . . . , sL are sols to Bezout’s identity s1q1 + . . .+ sLqL = 1.

Example 1

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Chinese Remainder Theorem for Commutative Rings

Let p1, . . . , pL be distinct primes, q = p1 · p2 . . . pL, and ql = q/pl

Z/qZ ∼= Zp1× Zp2

× . . .× ZpL∼= Fp1

× Fp2× . . .× FpL

An isomorphism:

M(v1, . . . , vL) = s1q1v1 + . . .+ sLqLv

L mod qZ,

where s1, . . . , sL are sols to Bezout’s identity s1q1 + . . .+ sLqL = 1.

Example 1

Consider Z/6Z ∼= F2 × F3. An isomorphism: M = 3v1 − 2v2 mod 6Z

1,0 1,20,10,21,10,0

1 2 3 50 4......

1,00,21,10,0

7 8 96

17 / 38


Construction πA (previously called product construction)

Let p1, . . . , pL be distinct primes and q = p1 p2 . . . pL.

Z/qZ ∼= Fp1× . . .× FpL

CL

C1

M

Fp1

FpL

Λ∗

+

ΠLl=1

plZN

Λ...

Product construction - L levels1 Choose (C1, . . . , CL) independently (no nesting) where Cl is over Fpl

18 / 38




Z/qZ ∼= Fp1× . . .× FpL

CL

C1

M

Fp1

FpL

Λ∗

+

ΠLl=1

plZN

Λ...


2 Λ∗ , M(C1, . . . , CL) where M is ring isomorphism

18 / 38




Z/qZ ∼= Fp1× . . .× FpL

CL

C1

M

Fp1

FpL

Λ∗

+

ΠLl=1

plZN

Λ...


2 Λ∗ , M(C1, . . . , CL) where M is ring isomorphism3 Λ , Λ∗ + qZN

4 λ ∈ Λ iff ϕ(λ) = (c1, . . . , cL) where ϕ , M−1 ◦ mod qZ

Works for other rings such as Z[i] and Z[ω] (and OK in general)

18 / 38




Z/qZ ∼= Fp1× . . .× FpL

CL

C1

M

Fp1

FpL

Λ∗

+

ΠLl=1

plZN

Λ...


2 Λ∗ , M(C1, . . . , CL) where M is ring isomorphism3 Λ , Λ∗ + qZN

4 λ ∈ Λ iff ϕ(λ) = (c1, . . . , cL) where ϕ , M−1 ◦ mod qZ

Works for other rings such as Z[i] and Z[ω] (and OK in general)

Reduce to Construction A when L = 118 / 38


Connection to Construction A with Coding over Ring

Construction A with Coding over Zq

• C linear code over Zq with a generator matrix G

• M : Zq → Z natural mapping

• Λ = M(C) + qZN = C + qZN

Construction πA is in fact a special case of this construction:

• q = p1 · p2 . . . · pL

• For z ∈ Zq, z = M(z1, . . . , zL) where zl ∈ Fplif z mod pl = zl

• G mod pl generates Cl for l ∈ {1, . . . , L}

This is an interesting special case that has connection to multilevelcoding/multistage decoding

19 / 38


Theorem 2

Exist Construction πA lattices that are good for channel coding under multistage

ML decoding

Proof.

• Follow the steps by Forney-Trott-Chung

• Modulo-qZN channel is symmetric (regular)

• Random multilevel linear codes achieve modulo-qZN channel capacity

• Let q = p1p2 . . . pL tend to ∞

Theorem 3

Exist Construction πA lattices that are good for MSE quantization

Proof.

• Follows the steps by Ordentlich-Erez

• Random multilevel linear codes induce uniform distribution over RN

• Let q = p1p2 . . . pL tend to ∞20 / 38


Power Constrained AWGN Channel: y = x+ z

Generalize Ordentlich-Erez’s construction to multilevel lattices:

Clc = {Gl

c ⊙wl|wl ∈ Fml

cpl }, Cl

f = {Glf ⊙wl|wl ∈ F

mlf

pl },

where Glf =

[

Glc Gl

]

,

Λf , γq−1M(C1f , . . . , C

Lf ) + γZN ,

Λc , γq−1M(C1c , . . . , C

Lc ) + γZN ,

• Clearly, Clc ⊂ Cl

f ; thus, Λc ⊂ Λf

• C = Λf ∩ Vol(VΛc) with R =

∑Ll=1

mlf−ml

c

Nlog(pl)

• Choose Λc good for MSE quantization and Λf good for coding

• Achieve AWGN capacity under multistage decoding

21 / 38


Low-complexity decoders

Serial modulo decoder (SMD):• Stage 1: Form estimate of c1 from

y1 = y mod p1Z

=(

M(c1, . . . , cL) + ΠLl=1plζ + z

)

mod p1Z

=(

c1 + z mod p1Z)

mod p1Z from CRT

• Stage s:Subtract all the contribution from the previous decoded stages to get

M(0, . . . , 0, cs, . . . , cL) + ΠLl=1plζ + z

Divide it by Πs−1

l=1pl to get

M(cs, . . . , cL) + ΠLl=splζ + z Construction πA with L− s+ 1 levels

where z , z/Πs−1

l=1pl

Form estimate of cs from

ys =

(

M(cs, . . . , cL) + ΠLl=splζ + z

)

mod psZ

= (cs + z mod psZ) mod psZ from CRT

22 / 38


Low-complexity decoders

Parallel modulo decoder (PMD):

• For s ∈ {1, . . . , L}, simultaneously form

ys = y mod psZ

=(

M(c1, . . . , cL) + ΠLl=1plζ + z

)

mod psZ

= (cs + z mod psZ) mod psZ from CRT

• Form estimate of cs from ys

• More loss but substantially lower latency

23 / 38


Extensions

Construction πD lattices:

• CRT only requires relatively prime rather than primes

• Allow all nature numbers: EX 12 = 4 · 3 hence Z/12Z ∼= Z4 × F3

• Coding over rings for those levels which do not happen to be fields

• Construction D is a special case with only 1 level Feng-Silva-Kschischang

• Can only show goodness for channel coding so far

Multilevel lattices over algebraic integers:

• Every OK forms a Dedekind domain

• Let I be ideal s.t. I = ΠLl=1pl

• CRT: OK/I ∼= OK/p1 × . . .×OK/pL ∼= Fpf11

× . . .× FpfLL

24 / 38

Application 1: Multistage Compute-and-Forward

Outline





5 Future Directions

25 / 38


Lattice codes and a modern view of interference

Nazer-Gastpar, Compute-and-forward: Harnessing interference through structuralcodes, T-IT 11

DM

uM

yM

zm

zM

+

z1

+

.

.

.

+

y1

ymxk

xK

h11

hMK

hm1

x1

w1

S1 D1

Sk Dm...

SK

u1

um

.

.

....

hmk

C

b b

b b b

b b

c1

wkC

b b

b b b

b b

ck

wKC

b b

b b b

b b

cK

· · ·

· · ·

• Source (Sk): Has message wk over Fp, where p is prime.

• Destination (Dm): ym =∑K

k=1 hmkxk + zm. No CSIT, only CSIR

• Recover um = ⊕Kk=1bmkwk where bmk ∈ Fp

• A building block of a large network

26 / 38


The Compute-and-Forward Paradigm

Theorem 4

Nazer-Gastpar For channel vector hm ∈ RK and integer vector am ∈ Z

K , the

following computation rate is achievable at Dm

R(hm, am) =1

2log+

(

1 + P‖hm‖2

‖am‖2 + P (‖hm‖2‖am‖2 − (hTmam)2)

)

How: To exploit the structural gains offered by the channel!

• Channel: ym =∑K

k=1 hmkxk + zm

• Sk: Use Construction A lattice code to match the channel structures tocertain extent

• Dm: Directly decode to a linear integer combination of codewords

• e.g.∑K

k=1 amkxk where amk ∈ Z

• Map this combination back to um = ⊕Kk=1bmk ⊙wk

27 / 38


Multistage Compute-and-Forward

Theorem 5

Same computation rate can be achieved with multistage decoding.

Proof.

• Split message into w1k × . . .×wL

k over Fp1× . . .× FpL

• Use the proposed multilevel lattices

• By CRT, uniquely represent amk = M(b1mk, . . . , bLmk) + qξ

K∑

k=1

amkxk =

K∑

k=1

[M(b1mk, . . . , bLmk) + qξ] · [M(c1k, . . . , c

Lk ) + qζ]

= M(

⊕Kk=1b

1mk ⊙ c1k, . . . ,⊕

Kk=1b

Lmk ⊙ cLk

)

+ qη

• Decoding can be done level by level without losing optimality

Substantially reduce decoding complexity28 / 38


Example of Multistage CF

DM

uM

yM

zm

zM

+

z1

+

.

.

.

+

y1

ymxk

xK

h11

hMK

hm1

x1

w1

S1 D1

Sk Dm...

SK

u1

um

.

.

....

hmk

C

b b

b b b

b b

c1

wkC

b b

b b b

b b

ck

wKC

b b

b b b

b b

cK

y = x1 + 5x2 + z

• Consider Z/6Z ∼= F2 × F3, same isomorphism

0 ↔ (0, 0), 1 ↔ (1, 1), 2 ↔ (0, 2),

3 ↔ (1, 0), 4 ↔ (0, 1), 5 ↔ (1, 2),

29 / 38


Example of Multistage CF

C2

C1

M

c11= 0

c21= 2

F2

F3

1,0 1,20,10,21,10,0

1 2 3 50 4...

1,00,21,10,0

7 8 96

0

2

x1 = M(0, 2) = 2

C2

C1

M

c12= 1

c22= 1

F2

F3

1,0 1,20,10,21,10,0

1 2 3 50 4...

1,00,21,10,0

7 8 96

1

1

x2 = M(1, 1) = 1

y = h1x1 + h2x2 = 7 = M(1, 1) + 6

h1 = 1 = M(1, 1)

h2 = 5 = M(1, 2)

1,0 1,20,10,21,10,0

1 2 3 50 4...

1,00,21,10,0

7 8 96

C2

C1

M

c11⊕ c1

2= 1

c21⊕ 2c2

2= 1

F2

1

1

30 / 38

Application 2: Lattice Index Coding

Outline





5 Future Directions

31 / 38


Broadcast channel with message side information

Natarajan-Hong-Viterbo T-IT 15

{w1, w2, . . . , wK}

noise

user 1

wS1

{w1, w2, . . . , wK}

...

noise

user L

wSL

{w1, w2, . . . , wK}

• Sender has independent messages {w1, w2, . . . , wK}.• Each receiver requests all the messages• Receiver l has wSl

a set of side info described by index set Sl

• For eg, S1 = {1, 2}, then wS1= {w1, w2}

• Noisy network yl = x+ zl where E[x2] ≤ P , and zl ∼ i.i.d. N (0, ρ2l )

32 / 38


Broadcast channel with message side information

Natarajan-Hong-Viterbo T-IT 15

{w1, w2, . . . , wK}

noise

user 1

wS1

{w1, w2, . . . , wK}

...

noise

user L

wSL

{w1, w2, . . . , wK}

• Sender has independent messages {w1, w2, . . . , wK}.• Each receiver requests all the messages• Receiver l has wSl

a set of side info described by index set Sl

• For eg, S1 = {1, 2}, then wS1= {w1, w2}

• Noisy network yl = x+ zl where E[x2] ≤ P , and zl ∼ i.i.d. N (0, ρ2l )

Noisy broadcasting problem with receiver side information

32 / 38


Capacity region and capacity-achieving codes

Capacity region (Tuncel T-IT 06): For every l ∈ {1, . . . , L},

1

2log2

(

1 +P

ρ2l

)

> H(w1, . . . , wK |wSl) =

K∑

k=1

Rk −RSl

• RSl=

∑

k∈SlRk

• Slepian-Wolf coding: Random codebooks + random binning + typicality

33 / 38




1

2log2

(

1 +P

ρ2l

)

> H(w1, . . . , wK |wSl) =

K∑

k=1

Rk −RSl

• RSl=

∑

k∈SlRk


A good code should translate every bit of side info into 6 dB SNR reduction

33 / 38




1

2log2

(

1 +P

ρ2l

)

> H(w1, . . . , wK |wSl) =

K∑

k=1

Rk −RSl

• RSl=

∑

k∈SlRk


A good code should translate every bit of side info into 6 dB SNR reduction

Lattice index codes by Natarajan-Hong-Viterbo T-IT 15

• Uniform side info gain of 6 dB by exploiting algebraic structure of CRT

• Extension to general ring of algebraic integers, Huang T-IT submitted 15

• Obtain diversity gains on top of side information gains

CRT seems to provide a right structure for this problem

33 / 38


Capacity-achieving lattice index codes

• Let wk ∈ Fpk, k ∈ {1, . . . ,K}

• Encode the messages by C = Λf ∩ Vol(VΛc)

Λf , γq−1M(C1f , . . . , C

kf , . . . , C

Kf ) + γZN ,

Λc , γq−1M(C1c , . . . , C

kc , . . . , C

Kc ) + γZN ,

• Here, M(v1, . . . , vL) , q1v1 + . . .+ qLv

L mod qZ

• Receiver l sees a codebook with messages in Sl fixed

Example 6 (3-User Case. S1 = {1}, S2 = {2, 3}, S3 = {1, 3})

Λf = γq−1M(C1f , C

2f , C

3f ) + γZN

Λc = γq−1M(C1c , C

2c , C

3c ) + γZN

34 / 38



• Let wk ∈ Fpk, k ∈ {1, . . . ,K}


Λf , γq−1M(C1f , . . . , C

kf , . . . , C

Kf ) + γZN ,

Λc , γq−1M(C1c , . . . , C

kc , . . . , C

Kc ) + γZN ,

• Here, M(v1, . . . , vL) , q1v1 + . . .+ qLv

L mod qZ




2f , C

3f ) + γZN


2c , C

3c ) + γZN

R2 +R3 ≤ 12 log(1 + P/ρ21) if Λf with C1

f fixed is good for coding

34 / 38



• Let wk ∈ Fpk, k ∈ {1, . . . ,K}


Λf , γq−1M(C1f , . . . , C

kf , . . . , C

Kf ) + γZN ,

Λc , γq−1M(C1c , . . . , C

kc , . . . , C

Kc ) + γZN ,

• Here, M(v1, . . . , vL) , q1v1 + . . .+ qLv

L mod qZ




2f , C

3f ) + γZN


2c , C

3c ) + γZN

R1 ≤ 12 log(1 + P/ρ22) if Λf with C2

f and C3f fixed is good for coding

34 / 38



• Let wk ∈ Fpk, k ∈ {1, . . . ,K}


Λf , γq−1M(C1f , . . . , C

kf , . . . , C

Kf ) + γZN ,

Λc , γq−1M(C1c , . . . , C

kc , . . . , C

Kc ) + γZN ,

• Here, M(v1, . . . , vL) , q1v1 + . . .+ qLv

L mod qZ




2f , C

3f ) + γZN


2c , C

3c ) + γZN

R2 ≤ 12 log(1 + P/ρ23) if Λf with C1

f and C3f fixed is good for coding

34 / 38



• Let wk ∈ Fpk, k ∈ {1, . . . ,K}


Λf , γq−1M(C1f , . . . , C

kf , . . . , C

Kf ) + γZN ,

Λc , γq−1M(C1c , . . . , C

kc , . . . , C

Kc ) + γZN ,

• Here, M(v1, . . . , vL) , q1v1 + . . .+ qLv

L mod qZ




2f , C

3f ) + γZN


2c , C

3c ) + γZN

In general, need good Construction πA lattices with arbitrary levels fixed

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Sketch of the proof

Consider 2 levels: Λ = p2C1 + p1C

2 + p1p2ZN . Note that

Λ = p1C2 + p2

(

C1 + p1ZN)

= p1C2 + p2Λ1

= p2C1 + p1

(

C2 + p2ZN)

= p2C1 + p1Λ2

• Randomly picking C1 results in good Λ1 w.h.p.


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Sketch of the proof

Consider 2 levels: Λ = p2C1 + p1C

2 + p1p2ZN . Note that

Λ = p1C2 + p2

(

C1 + p1ZN)

= p1C2 + p2Λ1

= p2C1 + p1

(

C2 + p2ZN)

= p2C1 + p1Λ2



• Note that p2Λ1 ⊂ Λ ⊂ Λ1

• Λ can be viewed as a Construction A lattice over base lattice Λ1

• Tailor (Loeliger’s version) Minkowski-Hlawka theorem specifically for thisconstruction

So picking C2 randomly results in good Λ w.h.p.

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Future Directions

Outline





5 Future Directions

36 / 38

Future Directions

Future Directions

Construction πA lattices:

Seeking for interesting problems where Construction πA can be useful

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Future Directions

Future Directions

Construction πA lattices:

Seeking for interesting problems where Construction πA can be useful

Number field lattices:

Construction A lattice codes over a general OK

• Kositwattanarerk-Ong-Oggier: Block fading wiretap channel

• Campello-Ling-Belfiore: Compound block fading channel

• Huang-Narayanan-Wang: Adaptive compute-and-forward

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Future Directions

CF over block fading channel

RX: [y(1)m , . . . ,y

(B)m ] where y

(b)m =

∑Kk=1 h

(b)mkx

(b)k + z

(b)m

• Construct lattice code from OK where K has degree B

• M : Fpf → OK/p ring isomorphism

• Λ = M(C) + pn and Λ = Ψ(Λ) where Ψ is canonical embedding

• λ = Ψ(λ) = [σ1(λ), . . . , σB(λ)]

• Enable computing

[

K∑

k=1

a(1)mkx

(1)k , . . . ,

K∑

k=1

a(B)mkx

(B)k

]

where Ψ−1([a(1)mk, . . . , a

(B)mk ]) ∈ OK

• Seem to better match channel’s structure than Z-lattices

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Future Directions

CF over block fading channel

RX: [y(1)m , . . . ,y

(B)m ] where y

(b)m =

∑Kk=1 h

(b)mkx

(b)k + z

(b)m

• Construct lattice code from OK where K has degree B

• M : Fpf → OK/p ring isomorphism

• Λ = M(C) + pn and Λ = Ψ(Λ) where Ψ is canonical embedding

• λ = Ψ(λ) = [σ1(λ), . . . , σB(λ)]

• Enable computing

[

K∑

k=1

a(1)mkx

(1)k , . . . ,

K∑

k=1

a(B)mkx

(B)k

]

where Ψ−1([a(1)mk, . . . , a

(B)mk ]) ∈ OK

• Seem to better match channel’s structure than Z-lattices

Good for coding has been proved by Campello-Ling-Belfiore. Need to show goodfor MSE quantization

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Chinese Remainder Theorem

Let p1, p2, . . . , pL be L distinct co-prime integers and q = p1.p2. . . . pL. Then,given

x ≡ a1(modp1)

x ≡ a2(modp2)

...

x ≡ aL(modpL)

there exists exactly one x ∈ Zq satisfying this system.

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Chinese Remainder Theorem

Let p1, p2, . . . , pL be L distinct co-prime integers and q = p1.p2. . . . pL. Then,given

x ≡ a1(modp1)

x ≡ a2(modp2)

...

x ≡ aL(modpL)

there exists exactly one x ∈ Zq satisfying this system.

EX: q = 6, p1 = 2, p2 = 3

x ≡ 0(mod2)

x ≡ 1(mod3)

x = 439 / 38

Modified M.-H. Theorem

Consider Λ = p2C1 + p1C

2 + p1p2Zn = p1C

2 + p2Λ1

• Let M2(v2) , p1v

2 mod p2Λ1 where v2 ∈ Fp2

• Let σ , M−1 ◦ mod p2Λ1

• Let C2 be the ensemble of (N, k2) linear codes over Fp2

1

|C2|

∑

C2∈C2

∑

v∈γΛ\0

f(v)

=1

|C2|

∑

C2∈C2

∑

v∈Λ1\0:σ(v)=0

f(γv) +∑

v∈Λ1\0:σ(v)∈C2\0

f(γv)

=∑

v∈Λ1\0:σ(v)=0

f(γv) +1

|C2|

∑

C2∈C2

∑

c∈C2\0

∑

v∈Λ1:σ(v)=c

f(γv)

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=∑

v∈Λ1\0:σ(v)=0

f(γv) +pk2

2 − 1

pN2 − 1

∑

c∈FNp2

∑

v∈Λ1:σ(v)=c

f(γv)

=∑

v∈Λ1\0:σ(v)=0

f(γv) +pk2

2 − 1

pN2 − 1

∑

v∈Λ1:σ(v) 6=0

f(γv)

(a)≈ pk2−N

2 γ−NVol(VΛ1)−1

∑

v∈γΛ1:σ(v) 6=0

f(v)γNVol(VΛ1)

(b)≈ Vol(VγΛ)

−1

∫

RN

f(v)dv,

• (a) requires p2 large and f(.) bounded

• (b) requires γNVol(VΛ1) small s.t. Riemann sum → Riemann integration

• Vol(VγΛ) = γNpk2−N2 Vol(VΛ1

)−1 = pk2−N2 pk1−N

1

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Construction πA Lattices: AReviewandRecent Results · 2019. 12. 20. · • Secrecy - He & Yener,...

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Transcript of Construction πA Lattices: AReviewandRecent Results · 2019. 12. 20. · • Secrecy - He & Yener,...