Low complexity normal bases

Abstract A normal basis in GF( q m ) is a basis of the form { β , β q , β q 2 ,…, β q m −1 }, i.e., a basis of conjugate elements in the field. In GF(2 m ) squaring with respect to a normal basis representation becomes simply a cyclic shift of the vector. For hardware design this is one of the very attractive features of these bases. Multiplication with respect to a normal basis can be defined in terms of a certain bilinear form. Define the complexity of the normal basis to the number of nonzero terms in this form. Again, for hardware design, it is important to find normal bases with low complexity. In this paper we investigate low complexity normal bases, give a construction for such bases and apply it to a number of cases of interest.