A Note on a Conjecture for Balanced Elementary Symmetric Boolean Functions
Abstract
In 2008, Cusick {\it et al.} conjectured that certain elementary symmetric Boolean functions of the form
σ
2
t+1
l−1,
2
t
are the only nonlinear balanced ones, where
t
,
l
are any positive integers, and
σ
n,d
=
⨁
1≤
i
1
<...<
i
d
≤n
x
i
1
x
i
2
...
x
i
d
for positive integers
n
,
1≤d≤n
. In this note, by analyzing the weight of
σ
n,
2
t
and
σ
n,d
, we prove that
wt(
σ
n,d
)<
2
n−1
holds in most cases, and so does the conjecture. According to the remainder of modulo 4, we also consider the weight of
σ
n,d
from two aspects: $n\equiv 3({\rm mod\}4)$ and $n\not\equiv 3({\rm mod\}4)$. Thus, we can simplify the conjecture. In particular, our results cover the most known results. In order to fully solve the conjecture, we also consider the weight of
σ
n,
2
t
+
2
s
and give some experiment results on it.