We study the (parameterized) complexity of a combinatorial problem, motivated by the desire to publish elections-related data, while preserving the privacy of the voters (humans or agents). In this problem, introduced and defined here, we are given an election, a voting rule, and a distance function over elections. The task is to find an election which is not too far away from the original election (with respect to the given distance function) while preserving the election winner (with respect to the given voting rule), and such that the resulting election is k-anonymized; an election is said to be k-anonymous if for each voter in it there are at least k − 1 other voters with the same preference order. We consider the problem of k-anonymizing elections for the Plurality rule and for the Condorcet rule, for the Discrete distance and for the Swap distance. We show that the parameterized complexity landscape of our problem is diverse, with cases ranging from being polynomial-time solvable to Para-NP-hard.