Parabolic Matrix Standard And Its Application, Under The Action Of Gl_n (f_q)

Let Fq be a finite field ,and charFg = p.The matrix A is called parabolic matrix if there is Ap = I for an element A of GLn(Fq).In the present paper ,we consider the problem of the normal form of the parabolic matrix in GLn(Fq).One construction of Cartetian authentication codes from the normal form of parabolic matrices over finite fields are presented and its size parameters are computed.Moreover,assume that the encoding rules are chosen according to a uniform probability distribution,the PI and PS,which devote the largest probabilities of a successful impersonation attack and of a successful substitution attack respectively,of these codes are also computed.