Paper 2005/399

On affine rank of spectrum support for plateaued function

Yuriy Tarannikov

Abstract

The plateaued functions have a big interest for the studying of bent functions and by the reason that many cryptographically important functions are plateaued. In this paper we study the possible values of the affine rank of spectrum support for plateaued functions. We consider for any positive integer $h$ plateaued functions with a spectrum support of cardinality $4^h$ (the cardinality must have such form), give the bounds on the affine rank for such functions and construct functions where the affine rank takes all integer values from $2h$ till $2^{h+1}-2$. We solve completely the problem for $h=2$, namely, we prove that the affine rank of any plateaued function with a spectrum support of cardinality $16$ is $4$, $5$ or $6$.

Note: This paper was submitted to WCC 2005 (March 14-18, 2005), accepted for the conference and revised according to the reviewer's propositions but it was not published in the proceedings since the author did not register for the conference. The paper probably will be published in Russian in the journal "Diskretnaya matematika" and in its English translation "Discrete mathematics and applications" in 2006. Nevertheless, this topic takes an interest (see, for example, the paper 2005/332 in this archive where some results were achieved using different approaches). Therefore, the author decided that it would be reasonable to maintain the paper in this archive.