线性递归序列生成的部分扩展类中的Bent函数
Bent Functions in the Partial Spread Class Generated by Linear Recurring Sequences
摘要：
提出了一种利用线性递归序列（LRS）生成的子空间构造部分扩展Bent函数的方法。我们首先证明了由两个LRS定义的线性映射的核有平凡交的充要条件是它们的反馈多项式是互质的。然后，我们的特点是适当的参数为一个家庭的成对互质多项式生成一个部分扩展所需的支持的弯曲功能，显示这样的家庭存在的当且仅当程度的基础多项式是$1$或$2$。然后，我们计算得到的多项式集，并证明，对于程度$1$，我们的LRS建设符合Deskiesian部分扩展。最后，我们执行计算机搜索的所有$\mathcal{PS}^-$和$\mathcal{PS}^+$弯曲功能的$n=8$变量产生我们的建设和计算其2-秩。结果表明，由$b=2$次数多项式定义的这些函数中的许多函数都不与任何Maiorana-McFarland或Deskiesian部分扩展函数EA等价.
英文摘要：
We present a construction of partial spread bent functions using subspaces generated by linear recurring sequences (LRS). We first show that the kernels of the linear mappings defined by two LRS have a trivial intersection if and only if their feedback polynomials are relatively prime. Then, we characterize the appropriate parameters for a family of pairwise coprime polynomials to generate a partial spread required for the support of a bent function, showing that such families exist if and only if the degrees of the underlying polynomials is either $1$ or $2$. We then count the resulting sets of polynomials and prove that for degree $1$, our LRS construction coincides with the Desarguesian partial spread. Finally, we perform a computer search of all $\mathcal{PS}^-$ and $\mathcal{PS}^+$ bent functions of $n=8$ variables generated by our construction and compute their 2-ranks. The results show that many of these functions defined by polynomials of degree $b=2$ are not EA-equivalent to any Maiorana-McFarland or Desarguesian partial spread function.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
