This paper proposes a method to construct a basis conversion matrix between two given bases in F_p^m. In the proposed method, Gauss period normal basis (GNB) works as a bridge between the two bases. The proposed method exploits this property and construct a basis conversion matrix mostly faster than EDF-based algorithm on average in polynomial time. Finally, simulation results are reported in which the proposed method compute a basis conversion matrix within 30 msec on average with Celeron (2.00GHz) when m log p ≈ 160.