Many efficient data structures use randomness, allowing them to improve upon deterministic ones. Usually, their efficiency and correctness are analyzed using probabilistic tools under the assumption that the inputs and queries are independent of the internal randomness of the data structure. In this work, we consider data structures in a more robust model, which we call the adversarial model. Roughly speaking, this model allows an adversary to choose inputs and queries adaptively according to previous responses. Specifically, we consider a data structure known as a "Bloom filter" and prove a tight connection between Bloom filters in this model and cryptography. A Bloom filter represents a set S of elements approximately by using fewer bits than a precise representation. The price for succinctness is allowing for some errors: For any x ∈ S, it should always answer Yes, and for any x S it should answer Yes only with small probability. In the adversarial model, we consider both efficient adversaries (that run in polynomial time) and computationally unbounded adversaries that are only bounded in the number of queries they can make. For computationally bounded adversaries, we show that non-trivial (memory-wise) Bloom filters exist if and only if one-way functions exist. For unbounded adversaries, we show that there exists a Bloom filter for sets of size n and error ϵ that is secure against t queries and uses only O(n log 1 ϵ + t ) bits of memory. In comparison, n log 1 ϵ is the best possible under a non-adaptive adversary.
Bibliographical noteFunding Information:
Supported in part by a grant from the I-CORE Program of the Planning and Budgeting Committee, the Israel Science Foundation, BSF, and the Israeli Ministry of Science and Technology. Moni Naor is the Incumbent of the Judith Kleeman Professorial Chair. Authors’ address: M. Naor and E. Yogev, Dept. Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, 76100 Israel; emails: firstname.lastname@example.org, email@example.com. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from firstname.lastname@example.org. © 2019 Association for Computing Machinery. 1549-6325/2019/05-ART35 $15.00 https://doi.org/10.1145/3306193
© 2019 Association for Computing Machinery.
- Adaptive inputs
- Bloom filter
- Pseudorandom functions
- Streaming algorithm