A note on the relation between XOR and Selective XOR Lemmas
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.
 Publication:

arXiv eprints
 Pub Date:
 April 2014
 arXiv:
 arXiv:1404.5169
 Bibcode:
 2014arXiv1404.5169J
 Keywords:

 Computer Science  Computational Complexity
 EPrint:
 The previous version has been significantly simplified to highlight the main result