Sphere Decoder Algorithm Lattice Reduction Algorithms Integer-Forcing Linear Receiver Lattice-based Cryptography Applications of Lattices in Telecommunications Amin Sakzad Dept of Electrical and Computer Systems Engineering Monash University amin.sakzad@monash.edu Oct. 2013 Lattice Coding III: Applications Amin Sakzad
Sphere Decoder Algorithm Lattice Reduction Algorithms Integer-Forcing Linear Receiver Lattice-based Cryptography Sphere Decoder Algorithm 1 Rotated Signal Constellations Sphere Decoding Algorithm Lattice Reduction Algorithms 2 Definitions Integer-Forcing Linear Receiver 3 Multiple-input Multiple-output Channel Problem statement Integer-Forcing Lattice-based Cryptography 4 GGH public-key cryptosystem Lattice Coding III: Applications Amin Sakzad
Sphere Decoder Algorithm Lattice Reduction Algorithms Integer-Forcing Linear Receiver Lattice-based Cryptography Rotated Signal Constellations Channel Model We consider n -dimensional signal constellation A carved from the lattice Λ with generator matrix G , for example 4 -QAM. Lattice Coding III: Applications Amin Sakzad
Sphere Decoder Algorithm Lattice Reduction Algorithms Integer-Forcing Linear Receiver Lattice-based Cryptography Rotated Signal Constellations Channel Model We consider n -dimensional signal constellation A carved from the lattice Λ with generator matrix G , for example 4 -QAM. Hence, x = uG represent a transmitted signal. Lattice Coding III: Applications Amin Sakzad
Sphere Decoder Algorithm Lattice Reduction Algorithms Integer-Forcing Linear Receiver Lattice-based Cryptography Rotated Signal Constellations Channel Model We consider n -dimensional signal constellation A carved from the lattice Λ with generator matrix G , for example 4 -QAM. Hence, x = uG represent a transmitted signal. The received vector y = α · x + z , where α i , are independent real Rayleigh random variables with unit second moment and z i are real Gaussian distributed with zero mean and variance σ/ 2 . Lattice Coding III: Applications Amin Sakzad
Sphere Decoder Algorithm Lattice Reduction Algorithms Integer-Forcing Linear Receiver Lattice-based Cryptography Rotated Signal Constellations Channel Model We consider n -dimensional signal constellation A carved from the lattice Λ with generator matrix G , for example 4 -QAM. Hence, x = uG represent a transmitted signal. The received vector y = α · x + z , where α i , are independent real Rayleigh random variables with unit second moment and z i are real Gaussian distributed with zero mean and variance σ/ 2 . With perfect Channel State Information (CSI) at the receiver, the ML decoder requires to solve the following optimization problem n � | y i − α i x i | 2 . min i =1 Lattice Coding III: Applications Amin Sakzad
Sphere Decoder Algorithm Lattice Reduction Algorithms Integer-Forcing Linear Receiver Lattice-based Cryptography Rotated Signal Constellations Pairwise error probability Using standard Chernoff bound technique one can estimate pairwise error probability under ML decoder as (4 σ ) ℓ Pr ( x → x ′ ) ≤ 1 4 σ � i ) 2 = min ,p ( x , x ′ ) 2 , ( x i − x ′ 2 d ( ℓ ) 2 x i � = x ′ i where the ℓ -product distance is d ( ℓ ) min ,p ( x , x ′ ) � � | x i − x ′ i | . x i � = x ′ i Lattice Coding III: Applications Amin Sakzad
Sphere Decoder Algorithm Lattice Reduction Algorithms Integer-Forcing Linear Receiver Lattice-based Cryptography Rotated Signal Constellations Goal Definition The parameter L = min( ℓ ) is called modulation diversity. Lattice Coding III: Applications Amin Sakzad
Sphere Decoder Algorithm Lattice Reduction Algorithms Integer-Forcing Linear Receiver Lattice-based Cryptography Rotated Signal Constellations Goal Definition The parameter L = min( ℓ ) is called modulation diversity. Definition We define the product distance as d min ,p = min d ( L ) min ,p . Lattice Coding III: Applications Amin Sakzad
Sphere Decoder Algorithm Lattice Reduction Algorithms Integer-Forcing Linear Receiver Lattice-based Cryptography Rotated Signal Constellations Goal Definition The parameter L = min( ℓ ) is called modulation diversity. Definition We define the product distance as d min ,p = min d ( L ) min ,p . To minimize the error probability, one should increase both L and d min ,p Lattice Coding III: Applications Amin Sakzad
Sphere Decoder Algorithm Lattice Reduction Algorithms Integer-Forcing Linear Receiver Lattice-based Cryptography Rotated Signal Constellations Rotated Z n -lattice constellations Lattice Coding III: Applications Amin Sakzad
Sphere Decoder Algorithm Lattice Reduction Algorithms Integer-Forcing Linear Receiver Lattice-based Cryptography Rotated Signal Constellations Rotated Z n -lattice constellations “Algebraic Number Theory” has been used as a strong tool to construct good lattices for signal constellations. Lattice Coding III: Applications Amin Sakzad
Sphere Decoder Algorithm Lattice Reduction Algorithms Integer-Forcing Linear Receiver Lattice-based Cryptography Rotated Signal Constellations Rotated Z n -lattice constellations “Algebraic Number Theory” has been used as a strong tool to construct good lattices for signal constellations. For these lattices, the minimum product distance will be related to the volume of the lattice and the “discriminant” of the underlying number field. Lattice Coding III: Applications Amin Sakzad
Sphere Decoder Algorithm Lattice Reduction Algorithms Integer-Forcing Linear Receiver Lattice-based Cryptography Rotated Signal Constellations Rotated Z n -lattice constellations “Algebraic Number Theory” has been used as a strong tool to construct good lattices for signal constellations. For these lattices, the minimum product distance will be related to the volume of the lattice and the “discriminant” of the underlying number field. The “signature” of a number field determines the modulation diversity. Lattice Coding III: Applications Amin Sakzad
Sphere Decoder Algorithm Lattice Reduction Algorithms Integer-Forcing Linear Receiver Lattice-based Cryptography Rotated Signal Constellations Rotated Z n -lattice constellations “Algebraic Number Theory” has been used as a strong tool to construct good lattices for signal constellations. For these lattices, the minimum product distance will be related to the volume of the lattice and the “discriminant” of the underlying number field. The “signature” of a number field determines the modulation diversity. List of good algebraic rotations are available online. See Emanuele’s webpage. Lattice Coding III: Applications Amin Sakzad
Sphere Decoder Algorithm Lattice Reduction Algorithms Integer-Forcing Linear Receiver Lattice-based Cryptography Sphere Decoding Algorithm Optimization Problem The problem is to solve the following: x ∈ Λ � y − x � 2 = w ∈ y − Λ � w � 2 . min min Lattice Coding III: Applications Amin Sakzad
Sphere Decoder Algorithm Lattice Reduction Algorithms Integer-Forcing Linear Receiver Lattice-based Cryptography Sphere Decoding Algorithm Algorithm[Viterbo’99] Set x = uG , y = ρ G , and w = ζ G for u ∈ Z n and ρ , ζ ∈ R n . Lattice Coding III: Applications Amin Sakzad
Sphere Decoder Algorithm Lattice Reduction Algorithms Integer-Forcing Linear Receiver Lattice-based Cryptography Sphere Decoding Algorithm Algorithm[Viterbo’99] Set x = uG , y = ρ G , and w = ζ G for u ∈ Z n and ρ , ζ ∈ R n . Let the Gram matrix M = GG T has the following Cholesky decomposition M = RR T , where R is an upper triangular matrix. Lattice Coding III: Applications Amin Sakzad
Sphere Decoder Algorithm Lattice Reduction Algorithms Integer-Forcing Linear Receiver Lattice-based Cryptography Sphere Decoding Algorithm Algorithm[Viterbo’99] Set x = uG , y = ρ G , and w = ζ G for u ∈ Z n and ρ , ζ ∈ R n . Let the Gram matrix M = GG T has the following Cholesky decomposition M = RR T , where R is an upper triangular matrix. We have n � w � 2 = ζ RR T ζ T = � q ii U 2 i ≤ C, i =1 where U i , q ii are based on r ij and ζ i , for 1 ≤ i, j ≤ n . Lattice Coding III: Applications Amin Sakzad
Sphere Decoder Algorithm Lattice Reduction Algorithms Integer-Forcing Linear Receiver Lattice-based Cryptography Sphere Decoding Algorithm Algorithm[Viterbo’99] Set x = uG , y = ρ G , and w = ζ G for u ∈ Z n and ρ , ζ ∈ R n . Let the Gram matrix M = GG T has the following Cholesky decomposition M = RR T , where R is an upper triangular matrix. We have n � w � 2 = ζ RR T ζ T = � q ii U 2 i ≤ C, i =1 where U i , q ii are based on r ij and ζ i , for 1 ≤ i, j ≤ n . Starting from U n and working backward, one can find bounds on U i , these will be transformed to bounds on u i . Lattice Coding III: Applications Amin Sakzad
Sphere Decoder Algorithm Lattice Reduction Algorithms Integer-Forcing Linear Receiver Lattice-based Cryptography Sphere Decoding Algorithm Comments The sphere decoding algorithm can be adapted to work on fading channels as well. Lattice Coding III: Applications Amin Sakzad
Sphere Decoder Algorithm Lattice Reduction Algorithms Integer-Forcing Linear Receiver Lattice-based Cryptography Sphere Decoding Algorithm Comments The sphere decoding algorithm can be adapted to work on fading channels as well. Choosing the radius C is a crucial part of the algorithm. Covering radius is an excellent choice. Lattice Coding III: Applications Amin Sakzad
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