output ${\widetilde{X}}$ or ${\widehat{X}}$, if the output is different from $X$ and also not $\bot$. This is only possible in the real implementation, and the probability of this is $\Pr\{{\widetilde{X}}\neq\bot\wedge{\widetilde{X}}\neq X\}$. This can also be studied through the trace distance between the two distributions $$\label{eq:trace-WCA}
    \begin{split}
      \delta(P_{XYY'{\widetilde{X}}},P_{XYY'{\widehat{X}}}) =
      \frac{1}{2}\sum_{m,y,y',x'}
      \Big|P_{XYY'{\widetilde{X}}}\big(m,y,y',x'\big)-
      P_{XYY'{\widehat{X}}}\big(m,y,y',x'\big)\Big|.
    \end{split}$$ Above, the index $x'$ runs over ${\mathcal{M}}\cup\{\bot\}$. Since the real and ideal cases are indistinguishable if $m=x'$, the above sum simplifies to the terms where $m\neq x'$. Furthermore, if $m\neq x'$ the ideal functionality ${\mathcal{F}}$ always outputs $\bot$. We can therefore change the name of the index to $m'$ since it now runs only over ${\mathcal{M}}$, and we find that the trace distance equals $\Pr\{{\widetilde{X}}\neq\bot\wedge{\widetilde{X}}\neq X\}$, because $$\label{eq:trace-WCA-2}
    \begin{split}
      \delta&(P_{XYY'{\widetilde{X}}},P_{XYY'{\widehat{X}}})= \sum_{m,y,y',m'\neq m}
      P_{XYY'{\widetilde{X}}}\big(m,y,y',m'\big)
      =\Pr\{{\widetilde{X}}\neq\bot\wedge{\widetilde{X}}\neq X\}\\
      &= \sum_{m,t,t',m'\neq m}       P_{X}(m)P_{Y|X}\big((m,t)|m\big)P_{Y'|XY}\big((m',t')\big|m,(m,t)\big)
      P_{{\widetilde{X}}|XYY'}\big(m'|m,(m,t),(m',t')\big)\\
      &= \sum_{m,t,t',m'\neq m}       P_{X}(m)\Pr\big\{h_K(m)=t\big\}P_{Y'|Y}\big((m',t')\big|(m,t)\big)
      \Pr\big\{h_K(m')=t'|h_K(m)=t\big\}\\
      &=\sum_{m,t,t',m'\neq m} P_{X}(m)       P_{Y'|Y}\big((m',t')\big|(m,t)\big)\Pr\{f_K(m')=t' \,\wedge\,
      f_K(m)=t\}.
    \end{split}$$ Now, the simple bound $\Pr\{f_K(m')=t'\,\wedge\,
  f_K(m)=t\}\le{\varepsilon}/|{\mathcal{T}}|+{\varepsilon}'$ (from Lemma \[lemma:subset\]) only gives $$\begin{split}
      \delta(P_{XYY'{\widetilde{X}}},P_{XYY'{\widehat{X}}})
      &=\sum_{m,t,t',m'\neq m} P_{X}(m)
      P_{Y'|Y}\big((m',t')|(m,t)\big)\Pr\{f_K(m')=t' \,\wedge\,
      f_K(m)=t\} \\
      &\le \sum_{m,t,t',m'\neq m}
      P_{X}(m) P_{Y'|Y}\big((m',t')|(m,t)\big)
      \Big(\frac{{\varepsilon}}{|{\mathcal{T}}|}+{\varepsilon}'\Big)={\varepsilon}+|{\mathcal{T}}|{\varepsilon}',
    \end{split}$$ and that is insufficient for our purposes. This occurs for the same reason as the high bounds in Theorems \[thm-WCA\] and \[thm-WCA-cor\]: the upper bound for the individual terms 