[Resource Topic] 2024/1103: A Note on Efficient Computation of the Multilinear Extension

Welcome to the resource topic for 2024/1103

Title:
A Note on Efficient Computation of the Multilinear Extension

Authors: Ron D. Rothblum

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.

ePrint: https://eprint.iacr.org/2024/1103

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