An Alternative Decoding Method for Gabidulin Codes in Characteristic Zero

Transcription

1 An Alternative Decoding Method for Gabidulin Codes in Characteristic Zero Sven Müelich, Sven Puchinger, David Mödinger, Martin Bossert Institute of Communications Engineering, Ulm University, Germany IEEE International Symposium on Information Theory (ISIT) Barcelona, July 10-15, 2016 An Alternative Decoding Method for Gabidulin Codes in Characteristic Zero (Sven Müelich, Sven Puchinger, David, Mödinger Martin Bossert) 1

2 Outline 1 Low-Rank Matrix Recovery 2 Gabidulin Codes in Characteristic Zero 3 Decoding An Alternative Decoding Method for Gabidulin Codes in Characteristic Zero (Sven Müelich, Sven Puchinger, David, Mödinger Martin Bossert) 2

3 Low-Rank Matrix Recovery (LRMR) Compressed Sensing LRMR b = Ax, b C m (known) A C m n (known) x C n sparse (unknown) (n m) Hamming Metric Coding Problem b = A(X), b C p (known) A : C m n C p linear (known) X C m n low rank (unknown) (p n m) Rank Metric Coding Problem s = H(c + e) = He s K n k syndrome (known) H K n k n pc matrix (known) c K n codeword (unknown) e K n, wt H(e) small (unknown) s = H(c + e) = He s L n k syndrome (known) H L n k n pc matrix (known) c L n codeword (unknown) e L n, wt R(e) small (unknown) (L n K m n ) An Alternative Decoding Method for Gabidulin Codes in Characteristic Zero (Sven Müelich, Sven Puchinger, David, Mödinger Martin Bossert) 4

4 Extension Fields L/K Galois extension (i.e. normal and separable), [L : K] =: m Galois group Gal (L/K) = {θ : L L automorphism s.t. θ(x) = x x K} Assumption: Gal (L/K) is cyclic, θ generator An Alternative Decoding Method for Gabidulin Codes in Characteristic Zero (Sven Müelich, Sven Puchinger, David, Mödinger Martin Bossert) 6

5 θ-polynomials L/K Galois extension, θ Gal (L/K). { d L[x; θ] = a = a ix i i=0 : a i L, d N } Addition (+) Multiplication ( ) a + b = (a i + b i)x i i a b = ( i a jθ j (b ) i j) x i (non-commutative) i j=0 Generalization of Linearized Polynomials Isomorphic to linearized polynomials in case F q m/f q, θ = q An Alternative Decoding Method for Gabidulin Codes in Characteristic Zero (Sven Müelich, Sven Puchinger, David, Mödinger Martin Bossert) 7

6 θ-polynomials (cont d) Evaluation a( ) := ev a : L L, α i aiθi (α) Degree deg a = max{i : a i 0} (max := ) Properties (a b)(α) = a(b(α)) deg(a b) = deg a + deg b Roots of a are subspace 1 of L a( ) : L L is an K-linear map 1 L is a K-vectorspace of dimension [L : K] Theorem (Augot, Loidreau, Robert) If Gal (L/K) is cyclic and θ generator dim ker(a) deg a An Alternative Decoding Method for Gabidulin Codes in Characteristic Zero (Sven Müelich, Sven Puchinger, David, Mödinger Martin Bossert) 8

7 Special θ-polynomials Annihilator Polynomial Subspace U L monic A U L[x; θ] of minimal degree: A U(u) = 0 u U Interpolation Polynomial x 1,..., x l L, linearly independent over K. y 1,..., y l L arbitrary. Then, I L[x; θ] of minimal degree: I(x i) = y i i = 1,..., l Theorem (Augot, Loidreau, Robert) If Gal (L/K) is cyclic and θ generator, deg A U = dim U (in general: deg A U dim U) deg I < l (and I is unique) An Alternative Decoding Method for Gabidulin Codes in Characteristic Zero (Sven Müelich, Sven Puchinger, David, Mödinger Martin Bossert) 9

8 Gabidulin Codes L/K Galois extension, θ Gal (L/K) generator. Definition g 1,..., g n L, linearly independent over K, k n m = [L : K] C G[n, k] = { c = [ f(g 1),..., f(g ] n) : f L[x; θ] deg f < k } L n extension c L n inverse extension C K m n Rank Metric: wt R(c) = rank(c), d R(c 1, c 2) = rank(c 1 C 2) Theorem (Augot, Loidreau, Robert) Minimum rank distance d = min c1 c 2 d R(c 1, c 2) = n k + 1 (MRD) Decoding: Augot, Loidreau, Robert (2013): O(n 3 ) Müelich, Puchinger, Mödinger, Bossert (2016): O(n 2 ) An Alternative Decoding Method for Gabidulin Codes in Characteristic Zero (Sven Müelich, Sven Puchinger, David, Mödinger Martin Bossert) 10

9 Decoding Error Model r = c + e L n, with wt R(e) =: τ Some Polynomials Λ := A e1,...,e n (unknown error span polynomial) ˆr interpolation polynomial with ˆr(g i) = r i i (known) Key Equation (Müelich, Puchinger, Mödinger, Bossert) Λˆr Λf mod A g1,...,g n Proof: ( Λ (ˆr f) ) (gi) = Λ(ˆr(g i) f(g i)) = Λ(r i c i) = Λ(e i) = 0 Thus, Λ (ˆr f) 0 mod A g1,...,g n characteristic zero equivalent to Gao s key equation for finite field Gabidulin codes (Wachter-Zeh) An Alternative Decoding Method for Gabidulin Codes in Characteristic Zero (Sven Müelich, Sven Puchinger, David, Mödinger Martin Bossert) 12

10 LSR Problem Key Equation (Müelich, Puchinger, Mödinger, Bossert) Λˆr Λf mod A g1,...,g n Linear Shift Register (LSR) Synthesis Problem Given ˆr, A g1,...,g n, find non-zero (λ, ω) L[x; θ] 2 with deg λ minimal and λˆr ω mod A g1,...,g n deg λ + k > deg ω Theorem (Müelich, Puchinger, Mödinger, Bossert) If τ = wt R(e) < d, the LSR has a solution (λ, ω) and for some α L, 2 (λ, ω) = α(λ, Λf) Assumption that θ is generator of Gal (L/K) necessary! An Alternative Decoding Method for Gabidulin Codes in Characteristic Zero (Sven Müelich, Sven Puchinger, David, Mödinger Martin Bossert) 13

11 Solving LSR Using Row Reduction Linear Shift Register (LSR) Synthesis Problem Given ˆr, A g1,...,g n, find non-zero (λ, ω) L[x; θ] 2 with deg λ minimal and λˆr ω mod A g1,...,g n deg λ + k > deg ω Row reduction ( is leading position = rightmost pos. of max. degree in row) [ x k ˆr ] 0 A g1,...,g n row operations [ m 11 x k m 12 m 21 x k m 22 ] ( ) = (λ, ω) = (m 11, m 12) Similar to the Extended Euclidean Algorithm (EEA) Advantage: Coefficient size reduction in intermediate computations ( ) Puchinger, Nielsen, Li, Sidorenko An Alternative Decoding Method for Gabidulin Codes in Characteristic Zero (Sven Müelich, Sven Puchinger, David, Mödinger Martin Bossert) 14

12 Decoding Summary Algorithm: Decode Gabidulin Codes Input: r = c + e Output: f s.t. c = [f(g 1),..., f(g n)] or decoding failure. 1 Calculate ˆr and A g1,...,g n 2 (λ, ω) Solve LSR with input ˆr, A g1,...,g n using row reduction 3 (Λ, Ω) α 1 (λ, ω) 4 (χ, ϱ) Right-divide Ω by Λ 5 if ϱ = 0 then return χ 6 else return decoding failure If wt R(e) < d, the algorithm finds f 2 Complexity/coefficient size growth tradeoff: Fast Normal Small growth Operations in L O(n 1.69 ) O(n 2 ) O(n 3 ) Row Reduction EEA [1] or PNLS [3] Fraction-free [4] [1] Wachter-Zeh 2013 [2] Puchinger, Müelich, Mödinger, Bossert 2016 [3] Puchinger, Nielsen, Li, Sidorenko 2015 [4] Beckermann, Cheng, Labahn 2006 D&Q [2] An Alternative Decoding Method for Gabidulin Codes in Characteristic Zero (Sven Müelich, Sven Puchinger, David, Mödinger Martin Bossert) 15

13 Thank you for your attention! An Alternative Decoding Method for Gabidulin Codes in Characteristic Zero (Sven Müelich, Sven Puchinger, David, Mödinger Martin Bossert) 16