We propose homomorphic algorithms for privacy-preserving applications where we are given an encrypted dataset and we want to compute the number of elements that share a common property. We consider a two party scenario between a client and a server, where the storage and computation is outsourced to the server. We present two new efficient methods to solve this problem by homomorphically evaluating a selection function encoding the desired property, and counting the number of elements which evaluates to the same value. Our first method programs the homomorphic computation in the style of the the functional bootstrapping of TFHE and can be instantiated with essentially any homomorphic encryption scheme that operates on polynomials, like FV or BGV. Our second method relies on new homomorphic operations and ciphertext formats, and it is more suitable for applications where the number of possible inputs is much larger than the number of possible values for the property. We illustrate the feasibility of our methods by presenting a publicly available proof-ofconcept implementation in C++ and using it to evaluate a heatmap function over encrypted geographic points.

It is known that fully homomorphic encryption (FHE) can be used to build efficient (labeled) Private Set Intersection protocols in the unbalanced setting, where one of the sets is much larger than the other~(Chen et al. (CCS'17, CCS'18)). In this paper we demonstrate multiple algorithmic improvements upon these works. In particular, our protocol has an asymptotically better computation cost, requiring only O(√|X| ) homomorphic multiplications, and communication complexity sublinear in the larger set size|X|. We demonstrate that our protocol is significantly better than that of Chen et al. (CCS'18) for many practical parameters, especially in terms of online communication cost. For example, when intersecting $228 and 2048 item sets, our protocol reduces the online computation time by more than 71% and communication by more than 63%. When intersecting 224 and 4096 item sets, our protocol reduces the online computation time by 27% and communication by 63%. Our comparison to other state-of-the-art unbalanced PSI protocols shows that our protocol has the best total communication complexity when |X| ≥ 224. For labeled PSI our protocol also outperforms Chen et al. (CCS'18). When intersecting 220 and 256 item sets, with the larger set having associated 288-byte labels, our protocol reduces the online computation time by more than 67% and communication by 34%. Finally, we demonstrate a modification that results in nearly constant communication cost in the larger set size |X|, but impractically high computation complexity on today's CPUs. For example, to intersect a 210-item set with sets of size 222, 224, or 226, our proof-of-concept implementation requires only 0.76 MB of online communication, which is more than a 24-fold improvement over Chen et al. (CCS'18).

Abstract Fully homomorphic encryption (FHE) allows to compute any function on encrypted values. However, in practice, there is no universal FHE scheme that is effi-cient in all possible use cases. In this work, we show that FHE schemes suitable for arithmetic circuits (e.g. BGV or BFV) have a similar performance as FHE schemes for non-arithmetic circuits (TFHE) in basic comparison tasks such as less-than, maximum and minimum operations. Our implementation of the less-than function in the HElib library is up to 3 times faster than the prior work based on BGV/BFV. It allows to compare a pair of 64-bit integers in 11 milliseconds, sort 64 32-bit integers in 19 seconds and find the minimum of 64 32-bit integers in 9.5 seconds on an average laptop without multi-threading.