I am trying to solve this problem that is about being able to de-anonymize a dataset. Can someone help with this point?: The adversary wants to de-anonymize one individual in the database, and learn something about their spending habits. Assume that for some row $r\in X^m$, the adversary gets access to auxiliary information $aux(r) \in (X \cup \{\bot\})^m$. This auxiliary information consists of information about $k$ transactions selected uniformly at random from the row $r$. Formally, we have that $r_i = aux_i$ for $i \in I$, and $aux_i = \bot$ for $i \notin I$, where the set of indices $I$ is chosen uniformly at random among all subsets of size $k$ of $[m]$.
Assume that $k > \frac{\log(\alpha/n)}{\log(1-\alpha)}$ for some $\alpha \in (0, 1)$.
Provide an algorithm that, given $aux(r)$ (with a uniformly random index set $I$) for a uniformly random $r \in D$, outputs a guess $\hat{r} \in X^m$ such that $\hat{r}$ and $r$ agree in at least a $(1-\alpha)$-fraction of the indices, with probability at least $1-\alpha$. This probability is taken over the random choice of $r$ and $I$, and the randomness (if any) of your algorithm.
Note: we have that: $$ \Pr[r'_i = \text{aux}(r)_i, \forall i \in I] \leq (1-\alpha)^k $$
for r' $\neq$ r
