A note on the relation between XOR and Selective XOR Lemmas

TL;DR

This paper establishes a reduction from the Selective XOR lemma to the standard XOR lemma, providing improved bounds and removing the need for sampling $(x, f(x))$ pairs, thus advancing understanding of unpredictability in Boolean functions.

Contribution

It introduces a reduction from the Selective XOR lemma to the standard XOR lemma with better bounds and no sampling assumptions.

Findings

Provides a quantitative reduction with improved bounds

Eliminates the need for sampling $(x, f(x))$ pairs

Enhances understanding of unpredictability in Boolean functions

Abstract

Given an unpredictable Boolean function $f: \{0, 1\}^n \rightarrow \{0, 1\}$, the standard Yao's XOR lemma is a statement about the unpredictability of computing $\oplus_{i \in [k]}f(x_i)$ given $x_1, ..., x_k \in \{0, 1\}^n$, whereas the Selective XOR lemma is a statement about the unpredictability of computing $\oplus_{i \in S}f(x_i)$ given $x_1, ..., x_k \in \{0, 1\}^n$ and $S \subseteq \{1, ..., k\}$. We give a reduction from the Selective XOR lemma to the standard XOR lemma. Our reduction gives better quantitative bounds for certain choice of parameters and does not require the assumption of being able to sample $(x, f(x))$ pairs.