Bloom Filters in Adversarial Environments

A Bloom filter represents a set $S$ of elements approximately, by using fewer bits than a precise representation. The price for succinctness is allowing some errors: for any $x \in S$ it should always answer 'Yes', and for any $x \notin S$ it should answer 'Yes' only with small probability. The latter are called false positive errors. In this work we study the possibility of Bloom filters in a stronger setting than the usual one, where an adversary can adaptively choose a sequence of queries according to previous results after the Bloom filter has been initialized. We consider both efficient adversaries (that run in polynomial-time) and computationally unbounded adversaries that are only bounded in the amount of queries they can make. We define a Bloom filter that maintains its error probability in this setting and say it is adversarial resilient. We show that non-trivial (memory-wise) adversarial resilient Bloom filters against computationally bounded adversaries exist if and only if one-way functions exist. The resulting construction is based on one-way permutations and is quite efficient and the proof of the necessity of one-way functions relies on the framework of adaptively changing distributions. For an unbounded adversary we show that there exist an $O(n\log{\frac{1}{\varepsilon}} + t)$ bits Bloom filter secure against $t$ queries (where $n$ is the set size and $\varepsilon$ is the false positive rate of the Bloom filter); in comparison, $n\log{\frac{1}{\varepsilon}}$ is the best possible under a non-adaptive adversary.