In this paper, we consider the k-approximate pattern matching problem under
differential privacy, where the goal is to report or count all substrings of a
given string S which have a Hamming distance at most k to a pattern P, or
decide whether such a substring exists. In our definition of privacy,
individual positions of the string S are protected. To be able to answer
queries under differential privacy, we allow some slack on k, i.e. we allow
reporting or counting substrings of S with a distance at most
(1+γ)k+α to P, for a multiplicative error γ and an
additive error α. We analyze which values of α and γ are
necessary or sufficient to solve the k-approximate pattern matching problem
while satisfying ϵ-differential privacy. Let n denote the length of
S. We give 1) an ϵ-differentially private algorithm with an additive
error of O(ϵ−1logn) and no multiplicative error for the existence
variant; 2) an ϵ-differentially private algorithm with an additive
error O(ϵ−1max(k,logn)⋅logn) for the counting variant; 3)
an ϵ-differentially private algorithm with an additive error of
O(ϵ−1logn) and multiplicative error O(1) for the reporting
variant for a special class of patterns. The error bounds hold with high
probability. All of these algorithms return a witness, that is, if there exists
a substring of S with distance at most k to P, then the algorithm returns
a substring of S with distance at most (1+γ)k+α to P. Further,
we complement these results by a lower bound, showing that any algorithm for
the existence variant which also returns a witness must have an additive error
of Ω(ϵ−1logn) with constant probability.Comment: This is a full version of a paper accepted to ITCS 202