Journal of the
Korean Mathematical Society JKMS

Motivated by XTR cryptosystem which is based on an irreducible polynomial $x^3 -cx^2 +c^px -1$ over ${F}_{p^2}$, we study polynomials of the form $F(c,x)=x^3 -cx^2 +c^qx -1$ over $F_{q^2}$ with $q=p^m$. In this paper, we establish a one to one correspondence between the set of such polynomials and a certain set of cubic polynomials over $F_q$. Our approach is rather theoretical and provides an efficient method to generate irreducible polynomials over $F_{q^2}$.

Keywords: irreducibility, normal basis, Hilbert Theorem 90

MSC numbers: 11T06, 11T55, 11T71