This paper proposes a useful method for constructing a self-dual normal basis in an arbitrary extension field F-p(m) such that 4p does not divide m(p - 1) and m is odd. In detail, when the characteristic p and extension degree in satisfies the following conditions (1) and either (2a) or (2b); (1) 2km + 1 is a prime number, (2a) the order of p in F2km+ 1 is 2km, (2b) 2 dagger km and the order of p in F2km + 1 is km, we can consider a class of Gauss period normal bases. Using this Gauss period normal basis, this paper shows a method to construct a self-dual normal basis in the extension field F-p(m). (C) 2008 Elsevier Inc. All rights reserved.