Computing all monomials of degree $n-1$ using $2n-3$ AND gates

TL;DR

This paper determines the exact number of AND gates needed to compute all degree (n-1) monomials in n variables, showing it is exactly 2n-3, which was previously unknown.

Contribution

It provides an AND-optimal implementation of the function, resolving the open problem of its precise multiplicative complexity.

Findings

Exact multiplicative complexity is 2n-3.

Provides an optimal implementation over AND, XOR, NOT gates.

Closes the gap between known bounds for the function.

Abstract

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 $2n-3$.