"There are 536 class of quartic forms Q (header) [in 8 boolean variables] providing bent functions of the form Q+f where f is a cubic functions." 
[Philippe Langevin, 2008.][1]

What is the current prospect of enumerating the [extended affine equivalence classes][2] of the bent functions of degree 4 in 8 boolean variables? Is the problem hopelessly large? Has anyone made an attack on it? Are the methods used by Fuller in [her thesis][2] still the state of the art? Could the methods used by Langevin et al. contribute to such an attack?




"Analysis of Affine Equivalent Boolean Functions for Cryptography, Joanne Fuller, 2003"


  [1]: http://langevin.univ-tln.fr/project/quartics/quartics.html "Classification of Boolean Quartics Forms in eight Variables"
  [2]: http://eprints.qut.edu.au/15828/1/Joanne_Fuller_Thesis.pdf