We consider the vector-valued Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}^n$ that outputs all $n$ monomials of degree $n-1$, i.e., $f_i(x)=\bigwedge_{j\neq i}x_j$, for $n\geq 3$. Boyar and Find have shown that the multiplicative complexity of this function is between $2n-3$ and $3n-6$. Determining its exact value has been an open problem that we address in this paper. We present an AND-optimal implementation of $f$ over the gate set $\{\text{AND},\text{XOR},\text{NOT}\}$, thus establishing that the multiplicative complexity of $f$ is exactly $2n-3$.

翻译：我们考虑向量值布尔函数 $f:\{0,1\}^n\rightarrow \{0,1\}^n$，该函数输出所有 $n$ 个 $n-1$ 次单项式，即对于 $n\geq 3$，有 $f_i(x)=\bigwedge_{j\neq i}x_j$。Boyar 和 Find 已证明该函数的乘法复杂度介于 $2n-3$ 与 $3n-6$ 之间。确定其确切值一直是本文所解决的开放问题。我们给出了在门集合 $\{\text{AND},\text{XOR},\text{NOT}\}$ 上 $f$ 的与门最优实现，从而确立了 $f$ 的乘法复杂度恰好为 $2n-3$。