Let C be a (n+t)x(n+t)
square matrix. Suppose each row of ciphertext matrix C is a ciphertext produced by the above basic encryption scheme under the (n+t)xt secret matrix S.
In general, the system of linear equations in max-plus algebra will have no solution, if A is
square matrix or if the number of columns in A is more than the number of rows in A.
The
square matrix A = {[a.sub.ij]}, i, j = [bar.1,n] is called an M-matrix, if it is non-singular, [a.sub.ij] [less than or equal to] 0, as i [not equal to] j and all the elements of matrix [A.sup.-1] are nonnegative (we denote it as [A.sup.-1] [greater than or equal to] 0).
One can be certain about the existence of the solution when
square matrix [K.sub.e] has one of the following properties [11]:
BA = ([d.sub.ij]) is a
square matrix of m order, where [d.sub.ij] = [[summation].sup.n.sub.k=1] [b.sub.ik] [a.sub.kj], i, j = 1, 2, ..., m:
(v) S is a
square matrix, of size m, which consists of diagonal values, each equal to the sum of rows, elements of matrix M, and zero elsewhere:
A complex reflection [tau] [member of] G(r, n) is associated with a
square matrix