Design criteria for space-time codes Union bound estimate of the error probability [Tarokh et al 1998] For a linear code, the difference of two codewords is still a codeword: � 1 P e ≤ (det( I + SNR XX † )) n X ∈C\{ 0 } � 1 ⇒ At high signal-to noise ratio ( SNR ), P e ≤ SNR nm (det( XX † )) n X ∈C\{ 0 } rank criterion: each nonzero codeword should be full-rank X ∈C\{ 0 } det( XX † ) determinant criterion: maximize inf Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 14
Design criteria for space-time codes Union bound estimate of the error probability [Tarokh et al 1998] For a linear code, the difference of two codewords is still a codeword: � 1 P e ≤ (det( I + SNR XX † )) n X ∈C\{ 0 } � 1 ⇒ At high signal-to noise ratio ( SNR ), P e ≤ SNR nm (det( XX † )) n X ∈C\{ 0 } rank criterion: each nonzero codeword should be full-rank X ∈C\{ 0 } det( XX † ) determinant criterion: maximize inf ⇒ the multiplicative structure of the code plays a role codes with non-vanishing determinant for any signal set achieve the DMT [Elia et al. 2006] Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 14
Space-time codes from cyclic division algebras F number field of degree k Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 15
Space-time codes from cyclic division algebras F number field of degree k K/F cyclic Galois extension of degree n , Gal( K/F ) = < σ > Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 15
Space-time codes from cyclic division algebras F number field of degree k K/F cyclic Galois extension of degree n , Gal( K/F ) = < σ > Cyclic algebra A = ( K/F, σ, γ ) = K ⊕ eK ⊕ · · · ⊕ e n − 1 K where e ∈ A satisfies the following properties: xe = eσ ( x ) ∀ x ∈ K , e n = γ ∈ F ∗ Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 15
Space-time codes from cyclic division algebras F number field of degree k K/F cyclic Galois extension of degree n , Gal( K/F ) = < σ > Cyclic algebra A = ( K/F, σ, γ ) = K ⊕ eK ⊕ · · · ⊕ e n − 1 K where e ∈ A satisfies the following properties: xe = eσ ( x ) ∀ x ∈ K , e n = γ ∈ F ∗ A is a division algebra if every nonzero element is invertible Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 15
Space-time codes from cyclic division algebras Left regular representation ψ : A → M n ( K ) ⊂ M n ( C ) a = x 0 + ex 1 + . . . + e n − 1 x n − 1 ∈ A γσ 2 ( x n − 2 ) γσ n − 1 ( x 1 ) x 0 γσ ( x n − 1 ) · · · γσ 2 ( x n − 1 ) γσ n − 1 ( x 2 ) x 1 σ ( x 0 ) σ 2 ( x 0 ) γσ n − 1 ( x 3 ) x 2 σ ( x 1 ) ψ ( a ) = . . ... . . . . σ 2 ( x n − 3 ) σ n − 1 ( x 0 ) x n − 1 σ ( x n − 2 ) · · · Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 16
Space-time codes from cyclic division algebras Left regular representation ψ : A → M n ( K ) ⊂ M n ( C ) a = x 0 + ex 1 + . . . + e n − 1 x n − 1 ∈ A γσ 2 ( x n − 2 ) γσ n − 1 ( x 1 ) x 0 γσ ( x n − 1 ) · · · γσ 2 ( x n − 1 ) γσ n − 1 ( x 2 ) x 1 σ ( x 0 ) σ 2 ( x 0 ) γσ n − 1 ( x 3 ) x 2 σ ( x 1 ) ψ ( a ) = . . ... . . . . σ 2 ( x n − 3 ) σ n − 1 ( x 0 ) x n − 1 σ ( x n − 2 ) · · · Obtain a matrix lattice Λ ⊂ M n ( C ) from a discrete subset of A : Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 16
Space-time codes from cyclic division algebras Left regular representation ψ : A → M n ( K ) ⊂ M n ( C ) a = x 0 + ex 1 + . . . + e n − 1 x n − 1 ∈ A γσ 2 ( x n − 2 ) γσ n − 1 ( x 1 ) x 0 γσ ( x n − 1 ) · · · γσ 2 ( x n − 1 ) γσ n − 1 ( x 2 ) x 1 σ ( x 0 ) σ 2 ( x 0 ) γσ n − 1 ( x 3 ) x 2 σ ( x 1 ) ψ ( a ) = . . ... . . . . σ 2 ( x n − 3 ) σ n − 1 ( x 0 ) x n − 1 σ ( x n − 2 ) · · · Obtain a matrix lattice Λ ⊂ M n ( C ) from a discrete subset of A : a subring O ⊂ A containing the identity is an order if it is a O F -module and generates A as a linear space over Q Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 16
Space-time codes from cyclic division algebras Left regular representation ψ : A → M n ( K ) ⊂ M n ( C ) a = x 0 + ex 1 + . . . + e n − 1 x n − 1 ∈ A γσ 2 ( x n − 2 ) γσ n − 1 ( x 1 ) x 0 γσ ( x n − 1 ) · · · γσ 2 ( x n − 1 ) γσ n − 1 ( x 2 ) x 1 σ ( x 0 ) σ 2 ( x 0 ) γσ n − 1 ( x 3 ) x 2 σ ( x 1 ) ψ ( a ) = . . ... . . . . σ 2 ( x n − 3 ) σ n − 1 ( x 0 ) x n − 1 σ ( x n − 2 ) · · · Obtain a matrix lattice Λ ⊂ M n ( C ) from a discrete subset of A : a subring O ⊂ A containing the identity is an order if it is a O F -module and generates A as a linear space over Q Λ = ψ ( O ) is a matrix lattice in M n ( C ) Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 16
Non-vanishing determinant property the determinant of the regular representation of an element is its reduced norm: det( ψ ( a )) = N A /F ( a ) � = 0 if a � = 0 Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 17
Non-vanishing determinant property the determinant of the regular representation of an element is its reduced norm: det( ψ ( a )) = N A /F ( a ) � = 0 if a � = 0 problem: the minimum determinant of the code C might vanish when |C| → ∞ Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 17
Non-vanishing determinant property the determinant of the regular representation of an element is its reduced norm: det( ψ ( a )) = N A /F ( a ) � = 0 if a � = 0 problem: the minimum determinant of the code C might vanish when |C| → ∞ Construction of NVD codes [Oggier et al. 2006], [Elia et al. 2006] if a ∈ Λ , N A /F ( a ) ∈ O F √ F = Q or Q ( − d ) ⇒ the ring of integers O F is discrete C ⊂ ψ ( O ) ⇒ X ∈C\{ 0 } | det X | ≥ 1 inf Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 17
Examples Alamouti code [Alamouti 1998] 2 transmit and 1 receive antenna, used in WiFi and 4G standards A is the algebra of Hamilton quaternions 1 � s 1 � − ¯ s 2 X = √ , s 1 , s 2 ∈ Z [ i ] s 2 s 1 ¯ 2 Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 18
Examples Alamouti code [Alamouti 1998] 2 transmit and 1 receive antenna, used in WiFi and 4G standards A is the algebra of Hamilton quaternions 1 � s 1 � − ¯ s 2 X = √ , s 1 , s 2 ∈ Z [ i ] s 2 s 1 ¯ 2 Golden Code [Belfiore et al 2005] 2 × 2 MIMO, optional profile in WiMAX standard A = ( Q ( i, θ ) / Q ( i ) , σ, i ) , θ golden number, α = 1 + iσ ( θ ) 1 � � α ( s 1 + s 2 θ ) α ( s 3 + s 4 θ ) X = √ , s 1 , s 2 , s 3 , s 4 ∈ Z [ i ] σ ( α ) i ( s 3 + s 4 σ ( θ )) σ ( α )( s 1 + s 2 σ ( θ )) 5 Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 18
Outline Coding for wireless communications 1 Single antenna systems MIMO systems Decoding 2 Algebraic reduction 3 Single antenna systems MIMO systems Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 19
Lattice point representation Example: the Golden Code � x 1 � � � 1 α ( s 1 + s 2 θ ) α ( s 3 + s 4 θ ) x 3 X = = √ αi ( s 3 + s 4 ¯ α ( s 1 + s 2 ¯ x 2 x 4 ¯ θ ) ¯ θ ) 5 0 0 x 1 α αθ s 1 α ¯ 1 x 2 0 0 αi ¯ ¯ θi s 2 x = v ( X ) = = = Φ s √ 0 0 x 3 α αθ s 3 5 α ¯ x 4 α ¯ ¯ θ 0 0 s 4 Vectorized system y = H l Φ s + w H l linear map corresponding to multiplication by H Φ (unitary) generator matrix s information vector Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 20
Lattice point representation Example: the Golden Code � x 1 � � � 1 α ( s 1 + s 2 θ ) α ( s 3 + s 4 θ ) x 3 X = = √ αi ( s 3 + s 4 ¯ α ( s 1 + s 2 ¯ x 2 x 4 ¯ θ ) ¯ θ ) 5 0 0 x 1 α αθ s 1 α ¯ 1 x 2 0 0 αi ¯ ¯ θi s 2 x = v ( X ) = = = Φ s √ 0 0 x 3 α αθ s 3 5 α ¯ x 4 α ¯ ¯ θ 0 0 s 4 Vectorized system y = H l Φ s + w H l linear map corresponding to multiplication by H Φ (unitary) generator matrix s information vector Maximum likelihood (ML) decoding Solve the closest vector problem (CVP) in the lattice generated by H l : � � y − H l x ′ � � 2 x = argmin ˆ x ′ ∈ v ( C ) Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 20
How hard are lattice problems in wireless communications? for general lattices, SVP and CVP are NP-hard [Ajtai 1998, Goldreich 1999] in lattice-based cryptography, average-case hardness is needed rather than worst-case hardness Ajtai discovered a connection between worst-case and average-case complexity of lattice problems Different notions of random lattices in mathematics: use the invariant measure on the space of lattices SL n ( R ) / SL n ( Z ) derived from the Haar measure on SL n ( R ) in cryptography: generator matrix is uniform mod q in communications: generator matrix has Gaussian entries Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 21
Decoding MIMO lattices ML decoders Sphere Decoder, Schnorr-Euchner algorithm... optimal performance but exponential complexity Suboptimal decoders zero forcing (ZF), successive interference cancellation (SIC)... polynomial complexity, but poor performance can be improved by preprocessing techniques Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 22
Sphere-decoding algorithm (Finkhe-Pohst) • • • • • • • • • • • • • • • • • • • • × × • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • enumerate all the lattice points inside a sphere centered in the received signal when a lattice point is found, the radius of the sphere can be updated apply a change of basis which maps the lattice into Z N : the sphere becomes an ellipsoid Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 23
Complexity of sphere decoding J. Jalden, B. Ottersten, “On the Complexity of Sphere Decoding in Digital Communications”, IEEE Transactions on Signal Processing vol 53 n.4, 2005 [Jaldén and Ottersten 2005]: the average complexity of the sphere decoding algorithm at fixed SNR is exponential and scales like L γN , where γ ∈ (0 , 1] depends on the SNR various techniques to reduce the complexity of sphere decoding: pruning of the decision tree, pre-processing, design of special fast-decodable codes... is it possible to achieve good performance with polynomial complexity? Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 24
Channel preprocessing Example: ZF decoding y = H x + w � H − 1 y � � x + H − 1 w � x ZF = ˆ = if H is orthogonal, ZF decoding is optimal if H is ill-conditioned, the noise H − 1 w is amplified Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 25
Channel preprocessing Example: ZF decoding y = H x + w � H − 1 y � � x + H − 1 w � x ZF = ˆ = if H is orthogonal, ZF decoding is optimal if H is ill-conditioned, the noise H − 1 w is amplified Solution: channel preprocessing by lattice reduction improves the performance of suboptimal decoders Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 25
Preprocessing using LLL reduction find a better lattice basis H red = HT, T unimodular Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 26
Preprocessing using LLL reduction find a better lattice basis H red = HT, T unimodular LLL-ZF decoder = LLL + Babai rounding compute the pseudo- inverse H † red �� �� H † x LLL − ZF = T ˆ red y Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 26
Preprocessing using LLL reduction find a better lattice basis H red = HT, T unimodular LLL-ZF decoder LLL-SIC decoder = LLL + Babai rounding = LLL + Babai nearest plane QR decomposition of H red compute the pseudo- y = Q H y = R x + Q H w inverse H † � red � � recursively compute yN ˜ x N = ˜ , rNN �� �� H † yi − � N x LLL − ZF = T ˆ � � j = i +1 rij ˜ xj ˜ red y x i = ˜ , i = N − 1 , . . . , 1 rii ˆ x LLL − SIC = T � x Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 26
Preprocessing using LLL reduction Complexity average number of iterations in the LLL algorithm for Rayleigh fading � � N 2 log N matrices ∼ O [Jalden et al. 2008] the worst-case number of iterations is unbounded each iteration requires O ( N 2 ) operations, which can be reduced to O ( N ) for LLL-SIC [Ling, Howgrave-Graham 2007] the average complexity of LLL-SIC is bounded by O ( N 3 log N ) Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 27
Preprocessing using LLL reduction Complexity average number of iterations in the LLL algorithm for Rayleigh fading � � N 2 log N matrices ∼ O [Jalden et al. 2008] the worst-case number of iterations is unbounded each iteration requires O ( N 2 ) operations, which can be reduced to O ( N ) for LLL-SIC [Ling, Howgrave-Graham 2007] the average complexity of LLL-SIC is bounded by O ( N 3 log N ) improved decoding techniques based on LLL: decoding by embedding [Luzzi, Rekaya, Belfiore 2010] , [Luzzi, Stehlé, Ling 2013] decoding by sampling [Liu, Ling, Stehlé 2011] , [Wang, Liu, Ling 2013] Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 27
Algebraic reduction up to now, algebraic tools have been used for coding but not for decoding Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 28
Algebraic reduction up to now, algebraic tools have been used for coding but not for decoding algebraic reduction is a right preprocessing method that exploits the multiplicative structure of the code Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 28
Algebraic reduction up to now, algebraic tools have been used for coding but not for decoding algebraic reduction is a right preprocessing method that exploits the multiplicative structure of the code main idea: absorb part of the channel into the code Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 28
Algebraic reduction up to now, algebraic tools have been used for coding but not for decoding algebraic reduction is a right preprocessing method that exploits the multiplicative structure of the code main idea: absorb part of the channel into the code approximate the channel matrix with a unit of the code Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 28
Outline Coding for wireless communications 1 Single antenna systems MIMO systems Decoding 2 Algebraic reduction 3 Single antenna systems MIMO systems Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 29
Outline Coding for wireless communications 1 Single antenna systems MIMO systems Decoding 2 Algebraic reduction 3 Single antenna systems MIMO systems Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 30
Algebraic reduction for fast fading channels G. Rekaya, J.-C. Belfiore, E. Viterbo, “A very efficient lattice reduction tool on fast fading channels”, ISITA 2004 Single antenna case: y = H x + w , x = ψ ( x ) ∈ Λ = ψ ( O K ) ideal lattice x = s 1 θ 1 + . . . + s n θ n ∈ O K Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 31
Algebraic reduction for fast fading channels G. Rekaya, J.-C. Belfiore, E. Viterbo, “A very efficient lattice reduction tool on fast fading channels”, ISITA 2004 Single antenna case: y = H x + w , x = ψ ( x ) ∈ Λ = ψ ( O K ) ideal lattice x = s 1 θ 1 + . . . + s n θ n ∈ O K Normalization of the received signal: y y ′ = = H 1 x + w ′ , det( H 1 ) = 1 � n det( H ) Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 31
Algebraic reduction for fast fading channels G. Rekaya, J.-C. Belfiore, E. Viterbo, “A very efficient lattice reduction tool on fast fading channels”, ISITA 2004 Single antenna case: y = H x + w , x = ψ ( x ) ∈ Λ = ψ ( O K ) ideal lattice x = s 1 θ 1 + . . . + s n θ n ∈ O K Normalization of the received signal: y y ′ = = H 1 x + w ′ , det( H 1 ) = 1 � n det( H ) Principle Approximate H 1 = diag( h ′ 1 , . . . , h ′ n ) with U l = diag( σ 1 ( u ) , σ 2 ( u ) , . . . , σ n ( u )) , where u is a unit of O K Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 31
The group of units O ∗ K Dirichlet’s unit theorem K algebraic number field with r 1 real Q -embeddings and 2 r 2 complex Q -embeddings, r = r 1 + r 2 − 1 . ∃ u 1 , . . . , u r fundamental units such that every u ∈ O ∗ K can be written as u = ζu e 1 1 · · · u e r r , where ζ ∈ R , the cyclic group of roots of unity in O K . Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 32
The group of units O ∗ K Dirichlet’s unit theorem K algebraic number field with r 1 real Q -embeddings and 2 r 2 complex Q -embeddings, r = r 1 + r 2 − 1 . ∃ u 1 , . . . , u r fundamental units such that every u ∈ O ∗ K can be written as u = ζu e 1 1 · · · u e r r , where ζ ∈ R , the cyclic group of roots of unity in O K . The logarithmic lattice Focus on the totally complex case: r 1 = 0 , r 2 = n . Consider f : O ∗ K → R n u �→ f ( u ) = (log | σ 1 ( u ) | , . . . , log | σ n ( u ) | ) Then f ( O ∗ K ) is an ( n − 1) -dimensional lattice in R n : � n � n i =1 | σ i ( x ) | 2 = N K/ Q ( x ) = 1 ⇒ i =1 log | σ i ( x ) | = 0 the volume of the logarithmic lattice depends on the regulator of the number field Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 32
Algebraic reduction for fast fading channels Approximate H 1 = diag( h ′ 1 , . . . , h ′ n ) with U l = diag( σ 1 ( u ) , σ 2 ( u ) , . . . , σ n ( u )) , where u is a unit of O K : H 1 = EU l , E = diag( e 1 , . . . , e n ) approximation error Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 33
Algebraic reduction for fast fading channels Approximate H 1 = diag( h ′ 1 , . . . , h ′ n ) with U l = diag( σ 1 ( u ) , σ 2 ( u ) , . . . , σ n ( u )) , where u is a unit of O K : H 1 = EU l , E = diag( e 1 , . . . , e n ) approximation error Units and unimodular transformations u unit of O K ⇔ U l Φ = Φ T u with T u unimodular (with entries in Z [ i ] ). ux ∈ O K ⇒ ux = � i s ′ i θ i Proof: U l ψ ( x ) = ψ ( ux ) = Φ s ′ = U l Φ s ⇒ s ′ = Φ − 1 U l Φ s , T u unimodular � �� � T u Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 33
Algebraic reduction for fast fading channels Approximate H 1 = diag( h ′ 1 , . . . , h ′ n ) with U l = diag( σ 1 ( u ) , σ 2 ( u ) , . . . , σ n ( u )) , where u is a unit of O K : H 1 = EU l , E = diag( e 1 , . . . , e n ) approximation error Units and unimodular transformations u unit of O K ⇔ U l Φ = Φ T u with T u unimodular (with entries in Z [ i ] ). ux ∈ O K ⇒ ux = � i s ′ i θ i Proof: U l ψ ( x ) = ψ ( ux ) = Φ s ′ = U l Φ s ⇒ s ′ = Φ − 1 U l Φ s , T u unimodular � �� � T u Received signal: y ′ = EU l Φ s + w ′ = E Φ T u s + w ′ = E Φ s ′ + w ′ , s ′ ∈ Z [ i ] n Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 33
Algebraic reduction for fast fading channels apply a suboptimal decoder (i.e. ZF): � Φ − 1 E − 1 y ′ � s ′ = s ′ + Φ − 1 E − 1 w ′ ˆ = ���� unitary Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 34
Algebraic reduction for fast fading channels apply a suboptimal decoder (i.e. ZF): � Φ − 1 E − 1 y ′ � s ′ = s ′ + Φ − 1 E − 1 w ′ ˆ = ���� unitary the i -th component of the equivalent noise is ( E − 1 w ′ ) i = σ i ( u ) i w ′ i h ′ � � � � � σ i ( u ) � should be small ∀ i = 1 , . . . , n to minimize noise variance, h ′ i ⇒ | log | σ i ( u ) | − log | h ′ i || should be small Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 34
Algebraic reduction for fast fading channels apply a suboptimal decoder (i.e. ZF): � Φ − 1 E − 1 y ′ � s ′ = s ′ + Φ − 1 E − 1 w ′ ˆ = ���� unitary the i -th component of the equivalent noise is ( E − 1 w ′ ) i = σ i ( u ) i w ′ i h ′ � � � � � σ i ( u ) � should be small ∀ i = 1 , . . . , n to minimize noise variance, h ′ i ⇒ | log | σ i ( u ) | − log | h ′ i || should be small How to find u ? find the closest point to (log | h ′ 1 | , . . . , log | h ′ n | ) in the logarithmic lattice. Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 34
Algebraic reduction for fast fading channels apply a suboptimal decoder (i.e. ZF): � Φ − 1 E − 1 y ′ � s ′ = s ′ + Φ − 1 E − 1 w ′ ˆ = ���� unitary the i -th component of the equivalent noise is ( E − 1 w ′ ) i = σ i ( u ) i w ′ i h ′ � � � � � σ i ( u ) � should be small ∀ i = 1 , . . . , n to minimize noise variance, h ′ i ⇒ | log | σ i ( u ) | − log | h ′ i || should be small How to find u ? find the closest point to (log | h ′ 1 | , . . . , log | h ′ n | ) in the logarithmic lattice. advantage: the logarithmic lattice is fixed once and for all and doesn’t depend on the channel Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 34
Algebraic reduction for fast fading channels algebraic reduction + ZF achieves the optimal diversity order it outperforms LLL + ZF in high dimension Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 35
Algebraic reduction for fast fading channels algebraic reduction + ZF achieves the optimal diversity order it outperforms LLL + ZF in high dimension Recent results used in [Campello, Ling, Belfiore 2017] to show that mod-p lattices achieve constant gap to compound capacity for n -antenna systems with reduced complexity Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 35
Algebraic reduction for fast fading channels algebraic reduction + ZF achieves the optimal diversity order it outperforms LLL + ZF in high dimension Recent results used in [Campello, Ling, Belfiore 2017] to show that mod-p lattices achieve constant gap to compound capacity for n -antenna systems with reduced complexity the performance depends on the covering radius r cov of the logarithmic lattice no known general bounds for r cov bounds for r cov in cyclotomic fields of prime power index [Cramer, Ducas, Peikert, Regev 2016] Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 35
Outline Coding for wireless communications 1 Single antenna systems MIMO systems Decoding 2 Algebraic reduction 3 Single antenna systems MIMO systems Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 36
Algebraic reduction for MIMO systems L. Luzzi, G. Rekaya - Ben Othman, J.-C. Belfiore, “Algebraic reduction for the Golden Code”, Adv. Math. Commun. 2012 Multiple antenna case: Y = HX + W A = ( K/ Q ( i ) , σ, γ ) division algebra, [ K : Q ( i )] = n X ∈ ψ ( O α ) , O maximal order of A Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 37
Algebraic reduction for MIMO systems L. Luzzi, G. Rekaya - Ben Othman, J.-C. Belfiore, “Algebraic reduction for the Golden Code”, Adv. Math. Commun. 2012 Multiple antenna case: Y = HX + W A = ( K/ Q ( i ) , σ, γ ) division algebra, [ K : Q ( i )] = n X ∈ ψ ( O α ) , O maximal order of A Normalization of the received signal: Y ′ = √ Y det( H ) Y ′ = H 1 X + W ′ , det( H 1 ) = 1 Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 37
Algebraic reduction for MIMO systems L. Luzzi, G. Rekaya - Ben Othman, J.-C. Belfiore, “Algebraic reduction for the Golden Code”, Adv. Math. Commun. 2012 Multiple antenna case: Y = HX + W A = ( K/ Q ( i ) , σ, γ ) division algebra, [ K : Q ( i )] = n X ∈ ψ ( O α ) , O maximal order of A Normalization of the received signal: Y ′ = √ Y det( H ) Y ′ = H 1 X + W ′ , det( H 1 ) = 1 Idea: approximate H 1 with a unit U ∈ O 1 Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 37
Algebraic reduction for MIMO systems H 1 = EU, E approximation error Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 38
Algebraic reduction for MIMO systems H 1 = EU, E approximation error in vectorized form: y ′ = E l U l Φ s + w ′ A l linear map corresponding to left multiplication by A Φ generator matrix of the code lattice s ∈ Z [ i ] N vector of QAM information signals, N = n 2 Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 38
Algebraic reduction for MIMO systems H 1 = EU, E approximation error in vectorized form: y ′ = E l U l Φ s + w ′ A l linear map corresponding to left multiplication by A Φ generator matrix of the code lattice s ∈ Z [ i ] N vector of QAM information signals, N = n 2 ⇔ U unit U l Φ = Φ T with T unimodular y ′ = E l Φ T s + w ′ = E l Φ s ′ + w ′ s ′ ∈ Z [ i ] N Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 38
Algebraic reduction for MIMO systems y ′ = E l Φ s ′ + w ′ s ′ ∈ Z [ i ] N Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 39
Algebraic reduction for MIMO systems y ′ = E l Φ s ′ + w ′ s ′ ∈ Z [ i ] N Apply ZF detection: � Φ − 1 E − 1 y ′ � � w ′ � s ′ = s ′ + Φ − 1 E − 1 ˆ = l Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 39
Algebraic reduction for MIMO systems y ′ = E l Φ s ′ + w ′ s ′ ∈ Z [ i ] N Apply ZF detection: � Φ − 1 E − 1 y ′ � � w ′ � s ′ = s ′ + Φ − 1 E − 1 ˆ = l the variance of the i -th noise component is bounded by Nσ 2 � Φ − 1 � � � E − 1 � � � 2 � 2 σ 2 i ≤ ∀ i = 1 , . . . , N 2 F | det( H ) | n Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 39
Algebraic reduction for MIMO systems y ′ = E l Φ s ′ + w ′ s ′ ∈ Z [ i ] N Apply ZF detection: � Φ − 1 E − 1 y ′ � � w ′ � s ′ = s ′ + Φ − 1 E − 1 ˆ = l the variance of the i -th noise component is bounded by Nσ 2 � � Φ − 1 � � E − 1 � � � 2 � 2 σ 2 i ≤ ∀ i = 1 , . . . , N 2 F | det( H ) | n How to choose U ? � � E − 1 � � � � � UH − 1 � ⇒ Choose U that minimizes F = 1 F Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 39
Quaternion case O 1 is a discrete subgroup Γ of SL 2 ( C ) Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 40
Quaternion case O 1 is a discrete subgroup Γ of SL 2 ( C ) � � H 1 U − 1 � � H 1 ∈ SL 2 ( C ) − → find U ∈ Γ s.t. � E � F = F is small Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 40
Quaternion case O 1 is a discrete subgroup Γ of SL 2 ( C ) � � H 1 U − 1 � � H 1 ∈ SL 2 ( C ) − → find U ∈ Γ s.t. � E � F = F is small Action of SL 2 ( C ) on hyperbolic 3 -space H 3 = { ( z, r ) | z ∈ C , r ∈ R + } with the hyperbolic distance ρ such that cosh ρ ( P, P ′ ) = 1 + d ( P,P ′ ) 2 rr ′ � a b � A = c d � Re( b ¯ | c | 2 + | d | 2 , Im( b ¯ d + a ¯ c ) d + a ¯ c ) 1 � J = (0 , 0 , 1) �→ A ( J ) = | c | 2 + | d | 2 , | c | 2 + | d | 2 Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 40
Quaternion case O 1 is a discrete subgroup Γ of SL 2 ( C ) � � H 1 U − 1 � � H 1 ∈ SL 2 ( C ) − → find U ∈ Γ s.t. � E � F = F is small Action of SL 2 ( C ) on hyperbolic 3 -space H 3 = { ( z, r ) | z ∈ C , r ∈ R + } with the hyperbolic distance ρ such that cosh ρ ( P, P ′ ) = 1 + d ( P,P ′ ) 2 rr ′ � a b � A = c d � Re( b ¯ | c | 2 + | d | 2 , Im( b ¯ d + a ¯ c ) d + a ¯ c ) 1 � J = (0 , 0 , 1) �→ A ( J ) = | c | 2 + | d | 2 , | c | 2 + | d | 2 � A � 2 F = 2 cosh ρ ( J, A ( J )) Relation to Frobenius norm: Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 40
Quaternion case O 1 is a discrete subgroup Γ of SL 2 ( C ) � � H 1 U − 1 � � H 1 ∈ SL 2 ( C ) − → find U ∈ Γ s.t. � E � F = F is small Action of SL 2 ( C ) on hyperbolic 3 -space H 3 = { ( z, r ) | z ∈ C , r ∈ R + } with the hyperbolic distance ρ such that cosh ρ ( P, P ′ ) = 1 + d ( P,P ′ ) 2 rr ′ � a b � A = c d � Re( b ¯ | c | 2 + | d | 2 , Im( b ¯ d + a ¯ c ) d + a ¯ c ) 1 � J = (0 , 0 , 1) �→ A ( J ) = | c | 2 + | d | 2 , | c | 2 + | d | 2 � A � 2 F = 2 cosh ρ ( J, A ( J )) Relation to Frobenius norm: � � H 1 U − 1 � � U − 1 ( J ) is close to H − 1 F is small ⇔ 1 ( J ) Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 40
Fundamental domain and generators of the group Poincaré’s polyhedron theorem the fundamental domain P for the action of Γ on H 3 is a compact hyperbolic polyhedron the copies v ( P ) , v ∈ Γ are isometric and form a tiling of H 3 there is a correspondence between a set of generators of the group and the set of side-pairings which map a face of P into another face v ( J ) Tamagawa volume formula v ( P ) 2 � Vol( P ) = ζ F (2) 3 J 4 π 2 | D F | ( N p − 1) P p | δ O Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 41
Discrete subgroups and fundamental domains Example: action of Z 2 on R 2 Action of Γ on H 3 the area enclosed by bisectors is a fundamental domain for the action the images of the fundamental domain form a tiling of R 2 Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 42
Discrete subgroups and fundamental domains Example: action of Z 2 on R 2 Action of Γ on H 3 the area enclosed by bisectors is a fundamental domain for the action the images of the fundamental domain form a tiling of R 2 Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 42
Discrete subgroups and fundamental domains Example: action of Z 2 on R 2 Action of à on H 3 the area enclosed by bisectors is a fundamental domain for the action ��������������������������� ��������������������������� the images of the fundamental domain form a tiling of R 2 Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 42
Discrete subgroups and fundamental domains Example: action of Z 2 on R 2 Action of Γ on H 3 the area enclosed by bisectors is a fundamental domain for the action the images of the fundamental domain form a tiling of R 2 Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 42
Discrete subgroups and fundamental domains Example: action of Z 2 on R 2 Action of Γ on H 3 the area enclosed by bisectors is a fundamental domain for the action the images of the fundamental domain form a tiling of R 2 Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 42
Discrete subgroups and fundamental domains Example: action of Z 2 on R 2 Action of à on H 3 the area enclosed by bisectors is a fundamental domain for the action �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� the images of the fundamental domain form a tiling of R 2 Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 42
Discrete subgroups and fundamental domains Example: action of Z 2 on R 2 Action of à on H 3 the area enclosed by the bisectors are Euclidean bisectors is a fundamental spheres domain for the action �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� the fundamental domain is a hyperbolic polyhedron the images of the the images of the fundamental domain form a fundamental domain form a tiling of H 3 tiling of R 2 Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 42
Intersecting bisectors: the Golden Code Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 43
Intersecting bisectors: the Golden Code Projection on the plane { r = 0 } Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 44
The fundamental polyhedron Projection on the plane { r = 0 } Laura Luzzi Algebraic reduction for lattice decoding Lattice Coding and Crypto Meeting 45
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