More answers for #2

Answering #2 from list of questions here:

An ONB of Type I exists a given field GF(2^{m}) if:

  • m+1 is a prime
  • 2 is a primitive in GF(m+1)

A Type II optimal normal basis exists in GF(2^{m}) if:

  • 2m+1 is prime
  • either 2 is a primitive in GF(2m+1) or 2m+1\equiv 3 \left (mod\; 4 \right ) and 2 generates the quadratic residues in GF(2m+1)

Interesting notes:

An ONB exists in GF(2^{m}) for 23% of all possible values of m

said this paper. Hmmm, that’s something.