基于Cayley图的平稳布尔函数的完全刻画
A complete characterization of plateaued Boolean functions in terms of their Cayley graphs
摘要：
在本文中，我们找到了一个完整的特征坪布尔函数的相关凯莱图。精确地说，我们证明了一个布尔函数$f$是$s$ -plateaued（重量$=2^{(n+s-2)/2}$）当且仅当相关的凯莱图是一个完全二部图之间的支持$f$和它的补充（因此，图是强正则的参数$e=0,d=2^{(n+s-2)/2}$）。此外，一个布尔函数$f$是$s$ -plateaued的（权值为$\neq 2^{(n+s-2)/2}$）当且仅当相关的Cayley图是强$3$ -行走正则的（对于所有奇数$\ell\geq 3$，也是强$\ell$ -行走正则的），具有一些显式给定的参数。
英文摘要：
In this paper we find a complete characterization of plateaued Boolean functions in terms of the associated Cayley graphs. Precisely, we show that a Boolean function $f$ is $s$-plateaued (of weight $=2^{(n+s-2)/2}$) if and only if the associated Cayley graph is a complete bipartite graph between the support of $f$ and its complement (hence the graph is strongly regular of parameters $e=0,d=2^{(n+s-2)/2}$). Moreover, a Boolean function $f$ is $s$-plateaued (of weight $\neq 2^{(n+s-2)/2}$) if and only if the associated Cayley graph is strongly $3$-walk-regular (and also strongly $\ell$-walk-regular, for all odd $\ell\geq 3$) with some explicitly given parameters.
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