Finding normal bases over finite fields with prescribed trace self-orthogonal relations [article]

Xiyong Zhang and Rongquan Feng and Qunying Liao and Xuhong Gao
2013 arXiv   pre-print
Normal bases and self-dual normal bases over finite fields have been found to be very useful in many fast arithmetic computations. It is well-known that there exists a self-dual normal basis of F_2^n over F_2 if and only if 4∤ n. In this paper, we prove there exists a normal element α of F_2^n over F_2 corresponding to a prescribed vector a=(a_0,a_1,...,a_n-1)∈F_2^n such that a_i=Tr_2^n|2(α^1+2^i) for 0≤ i≤ n-1, where n is a 2-power or odd, if and only if the given vector a is symmetric
more » ... -i for all i, 1≤ i≤ n-1), and one of the following is true. 1) n=2^s≥ 4, a_0=1, a_n/2=0, ∑_1≤ i≤ n/2-1, (i,2)=1a_i=1; 2) n is odd, (∑_0≤ i≤ n-1a_ix^i,x^n-1)=1. Furthermore we give an algorithm to obtain normal elements corresponding to prescribed vectors in the above two cases. For a general positive integer n with 4|n, some necessary conditions for a vector to be the corresponding vector of a normal element of F_2^n over F_2 are given. And for all n with 4|n, we prove that there exists a normal element of F_2^n over F_2 such that the Hamming weight of its corresponding vector is 3, which is the lowest possible Hamming weight.
arXiv:1303.2283v1 fatcat:4kh5b2pwjnd23jeqtl53mu5mey