**ON THE DEGREE OF HOMOGENEOUS BENT FUNCTIONS**

*Qingshu Meng and Huanguo Zhang and Min Yang and Jingsong Cui*

**Abstract: **It is well known that the degree of a $2m$-variable bent function
is at most $m.$ However, the case in homogeneous bent functions is
not clear. In this paper, it is proved that there is no
homogeneous bent functions of degree $m$ in $2m$ variables when
$m>3;$ there is no homogenous bent function of degree $m-1$ in 2m
variables when $m>4;$ Generally, for any nonnegative integer $k$,
there exists a positive integer $N$ such that when $m>N$, there is
no homogeneous bent functions of degree $m-k$ in $2m$ variables.
In other words, we get a tighter upper bound on the degree of
homogeneous bent functions. A conjecture is proposed that for any
positive integer $k>1$, there exists a positive integer $N$ such
that when $m>N$, there exists homogeneous bent function of degree
$k$ in $2m$ variables.

**Category / Keywords: **secret-key cryptography / bent functions, Walsh transform, algebraic degree

**Date: **received 1 Nov 2004, last revised 1 Dec 2005

**Contact author: **mqseagle at sohu com

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

**Note: **a scholar told me one spelling error due to my unfamilarity to latex when i submitted this paper to eprint in 2004.

**Version: **20051201:071440 (All versions of this report)

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