Paper 2024/1103

A Note on Efficient Computation of the Multilinear Extension

Abstract

The multilinear extension of an $m$-variate function $f : \{0,1\}^m \to \mathbb{F}$, relative to a finite field $\mathbb{F}$, is the unique multilinear polynomial $\hat{f} : \mathbb{F}^m \to \mathbb{F}$ that agrees with $f$ on inputs in $\{0,1\}^m$. In this note we show how, given oracle access to $f : \{0,1\}^m \to \mathbb{F}$ and a point $z \in \mathbb{F}^m$, to compute $\hat{f}(z)$ using exactly $2^{m+1}$ multiplications, $2^m$ additions and $O(m)$ additional operations. The amount of space used corresponds to $O(m)$ field elements.