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We give a strong direct sum theorem for computing $xor \circ g$.Specifically, we show that for every function g and every $k\geq 2$, therandomized query complexity of computing the xor of k instances of g satisfies$\overline{R}_\eps(xor\circ g) = \Theta(k \overline{R}_{\eps/k}(g))$. Thismatches the naive success amplification upper bound and answers a conjecture ofBlais and Brody (CCC19).  As a consequence of our strong direct sum theorem, we give a total function gfor which $R(xor \circ g) = \Theta(k \log(k)\cdot R(g))$, answering an openquestion from Ben-David et al.(arxiv:2006.10957v1).

A Strong XOR Lemma for Randomized Query Complexity