**Unbalanced Elementary Symmetric Boolean Functions with the Degree "d" and "wt(d)>=3"**

*Zhihui Ou*

**Abstract: **In the paper, for $d=2^{t}k$, $n=2^{t}(2k+q)+m$ and special $k=2^{w}(2^0+2^1+\cdots+2^s)$, we present that a majority of $X(d,n)$ are not balanced. The results include many cases $wt(d)\geq 3$ and $n\equiv 0,1,2,3 mod4$. The results are also parts of the conjecture that $X(2^t,2^{t+1}l-1)$ is only nonlinear balanced elementary symmetric Boolean function. Where $t\geq 2$, $q\geq 1$, $s\geq 0$, $w\geq 0$ and $m\geq -1$ are integers, and $X(d,n)=\bigoplus\limits_{1\le i_{1} <\cdots<i_{d}\le n} x_{i_{1} } \cdots x_{i_{d} }$.\newline\newline

**Category / Keywords: **Cryptograph, Boolean functions, balancedness, elementary symmetric.

**Date: **received 24 Feb 2012, last revised 29 Feb 2012, withdrawn 23 Apr 2012

**Contact author: **good_0501_oudi at 163 com

**Available format(s): **(-- withdrawn --)

**Version: **20120423:103658 (All versions of this report)

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