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### Paper:

TR19-105 | 16th August 2019 11:08

#### A note on the relation between XOR and Selective XOR Lemmas

TR19-105
Authors: Ragesh Jaiswal
Publication: 16th August 2019 11:16
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.