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On the Efficiency of FHE-based Private Queries

On the Efficiency of FHE-based Private Queries


Private query processing is a very attractive problem in the fields of both cryptography and databases. In this work, we restrict our attention to the efficiency aspect of the problem, particularly for basic queries with conditions on various combinations of equality. Without loss of generality, these conditions can be regarded as a Boolean function, and this Boolean function can then be evaluated at ciphertexts produced by a fully homomorphic encryption (FHE) scheme without decryption. From the efficiency perspective, the remaining concern is to efficiently test the equality function without severely downgrading the performance of FHE-based querying solutions.

To this end, we first analyze the multiplicative depth required for an equality test algorithm with respect to the plaintext space inhabited by general FHE schemes. The primary reason for this approach is that given an equality test algorithm, its efficiency is measured in terms of the multiplicative depth required to construct its arithmetic circuit expression. Indeed, the implemented equality test algorithm dominates the entire performance of FHE-based query solutions, apart from the performance of the underlying FHE scheme. Then, we measure the multiplicative depth considering an FHE scheme that takes an extension field as its plaintext space and that supports the depth-free evaluation of Frobenius maps. According to our analysis, when the plaintext space of an FHE scheme is a field of characteristic 2, the equality test algorithm for `-bit messages requires the lowest multiplicative depth dlog `e. Furthermore, we design a set of private query protocols for conjunctive, disjunctive, and threshold queries based on the equality test algorithm. Similarly, applying the equality test algorithm over F2` , our querying protocols require the minimum depths. More specifically, a multiplicative depth of dlog `e + dlog (1 + _)e is required for conjunctive and disjunctive queries, and a depth of dlog `e + 2dlog (1 + _)e is required for threshold conjunctive queries, when their query conditions have _ attributes to be compared. Finally, we provide a communication-efficient version of our solutions, though with additional computational costs, when an upper bound _ (0 _ _ _ 1) on the selectivity of a database is given. Consequently, we reduce the communication cost from n to approximately b_nc ciphertexts with dlog ne additional depth when the database consists of n tuples.

Existing System:

Performance-critical systems, we recommend somewhat homomorphic encryption (SHE) if possible, but if circumstances do not permit this approach, an efficient leveled FHE scheme should be considered while minimizing the multiplicative depth of the target algorithms. Because an equality test algorithm can take as input a large plaintext message, the minimum requirement for the implementation of a private query system is to use an efficient leveled FHE scheme with depth-free automorphisms, and further optimization is recommended.


Proposed System:

In principle our proposed protocols work by repeatedly invoking an equality test algorithm as a sub-routine. Different from our choice, one may implement an algorithm for equality test based on SHE. All currently existing SHE and (leveled) FHE schemes are constructed by following Gentry’s strategy that is to insert small noise into a ciphertext and so noise in a result of the evaluation algorithm grows up. While the noise growth in SHE schemes is exponential in the number of multiplications, that in leveled FHE schemes is polynomial. Thus a leveled FHE is a better choice for efficient  implementations of our protocols.

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