The public-key is G' = SGP, where G is a k x n generator matrix for the Goppa code, S is a k x k non-singular matrix, and P is an n x n permutation matrix
. In addition, the private keys are S, G, and P.
, [j.sub.#J]} yields a permutation matrix
[P.sub.J] with entries [p.sub.ij] = 0 for i,j = 1, ..., n, except [mathematical expression not reproducible] for i = 1, ...
It can be shown that the optimal linear processing matrix of the special "relay" is a promotion of the permutation matrix
. To maximize the system throughput under fixed power allocation, the optimal linear processing matrix should be found out.
[PI] is a permutation matrix
. [[DELTA].sub.i], i =1, ..., 3, are diagonal scaling matrices satisfying [[DELTA].sub.1] [[DELTA].sub.2] [[DELTA].sub.3] = [I.sub.K]
(vi) An n x n matrix is called a permutation matrix
if it is formed from the identity matrix by reordering its columns and/or rows.
Zhang, "Reusing the permutation matrix
dynamically for efficient image cryptographic algorithm," Signal Processing, vol.
A matrix A is called reducible if there exists an n order permutation matrix
P such that
Two pivot orderings O, O' [member of] O([P.sub.m]) are permutation equivalent if [M.sub.O'] = P[M.sub.O][P.sup.T] holds for some permutation matrix
According to the sequence [B.sub.1] produced by the Chen system, the permutation index sequence X is obtained in ascending order, and X is populated according to the M value of each line to obtain the permutation matrix
T, which is used for the position scrambling of the image pixels.
where [j.sup.2] = -1, [(*).sup.#] denotes pseudoinverse matrix, P is a permutation matrix
(corresponding to an arbitrary order of restitution of the sources), [DELTA] is a diagonal matrix (corresponding to arbitrary scaling for the recovered sources), [PHI] = [[[[phi].sub.1], ..., [[phi].sub.n]].sup.T], [for all][[phi].sub.i] [member of] R represents the phase vector (corresponding to phase shift ambiguity in complex domain of the source signals), and diag(a) is square diagonal matrix containing the elements of the vector a.
Then we have to reorder equations [PHI] := [[[[phi].sub.1] [[phi].sub.2]].sup.T] with respect to the row degrees starting from the lowest that can be done by means of multiplication by the permutation matrix