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Example text
Iκ , {i1 , . . , iκ } = N (y ( ) ), we denote it in slightly misusing the notation by V N (y( ) ) . Let V −1 be the inverse of N (y ( ) ) V N (y ( ) ) , then we can use the matrix ⎛ −1 ⎞ V N (y(1) ) 0 ... 0 ⎜ ⎟ .. ⎜ ⎟ . 0 V −1 . ⎜ ⎟ (2) ) N (y U =⎜ ⎟ .. ⎜ .. ⎟ . ⎝ ⎠ . 0 −1 0 ... 0 V N (y (l) ) to calculate ⎛ I 0 ... 0 ⎜ ⎜0 0 ... 0 ⎜ ⎜0 I ... 0 ⎜ ˜ =U A =⎜ A ⎜0 0 ... 0 ⎜. . ⎜ . .. . ⎜. ⎜ ⎝0 0 ... I 0 0 ... 0 (1) −W 1 · L (1) −W 2 · L (2) −W 1 · L (2) −W 2 · L .. (l) −W 1 · L (l) −W 2 · L ⎛ (1) T⎞ W 1 · y (1) ⎜ (1) (1) T ⎟ (1) ⎟ ⎜W 2 · y ⎟ ⎟ ⎜ ⎟ (2) ⎟ ⎜ (2) (2) T ⎟ ⎟ W · y ⎜ ⎟ ⎟ ⎜ 1(2) (2) T ⎟ (2) ⎟ ˜ ⎜ ⎟, ⎟ , and b = U b = ⎜W 2 · y ⎟ ⎟ ⎜ ⎟ .
Nk+t−1 )T where the ni are the coefficients of the q-polynomial N . Set ⎛ ⎞⎫ [k+t−1] [t] g1 · · · g1 y1 · · · y1 ⎪ ⎪ ⎬ ⎜ . .. . .. ⎟ ⎟ . S=⎜ . . ⎠⎪ n ⎝ . ⎪ [k+t−1] [t] ⎭ g ··· g y ··· y n n n n Solving (2) is equivalent to solving the system S× N V = 0. (3) In the unknowns N and V. Therefore it costs roughly (k + 2t)3 operations over GF (q). It is far too much to be efficiently implemented, compared to the already existing decoding algorithms. By considering (3), it is clear that a part of the matrix S is independent of the received word, depending only on the parameters of the Gabidulin code.
Ideals over a noncommutative ring and their application in cryptology. In D. W. Davies, editor, Advances in Cryptology – EUROCRYPT’91, volume 547 of LNCS, pages 482–489. Springer-Verlag, 1991. 6. P. Loidreau. Sur la reconstruction des polynˆ omes lin´eaires : un nouvel algorithme de d´ecodage des codes de Gabidulin. Comptes Rendus de l’Acad´ emie des Sciences: S´erie I, 339(10):745–750, 2004. 7. Ø. Ore. On a special class of polynomials. Transactions of the American Mathematical Society, 35:559–584, 1933.