In this paper, we introduce POPPY, a novel suite of privacy-preserving algorithms designed for computing spectral centrality measures over graphs distributed across mutually distrustful data centers. POPPY is the first approach that achieves together generality, accuracy, and scalability with respect to the number of participants. POPPY uses the CKKS fully homomorphic encryption scheme to support the arithmetic division operation over ciphertexts. POPPY consists of three variants: POPPYs, optimized for sparse graphs using SIMD operations; POPPYd, tailored for dense graphs through efficient encrypted matrix-vector multiplication; and POPPYh, a hybrid between POPPYs and POPPYd for coping with contexts involving a large number of remote nodes. In addition to its core algorithms, POPPY comes with a pruning strategy based on a new notion of node equivalence called INC-equivalence. Pruning is indeed of utmost importance when using fully homomorphic encryption in order to minimize the number of encrypted operations performed. The INC-equivalence notion allows us to eliminate redundant nodes efficiently and without any impact on the accuracy of the centrality scores. Our comprehensive theoretical analysis and empirical evaluation on real-world and synthetic datasets demonstrate that POPPY achieves together generality, accuracy, and scalability.

Cryptographic approaches, such as secure multiparty computation, can be used to securely compute a function of a distributed graph without centralizing the data of each participant. However, the output of the protocol can leak sensitive information about the structure of the original graph. In particular, we propose an approach by which an adversary observing the result of a private protocol for the computation of the number of common neighbors between all pairs of vertices, can reconstruct the adjacency matrix of the graph. In fact, this can only be done up to co-squareness, a notion we introduce, as two different graphs can have the same matrix of common neighbors. To realize this, we consider two adversary models, one who observes the common neighbors matrix only and a more informed one that has partial knowledge of the original graph. Our results demonstrate that, from their common neighbors matrix, graphs can be reconstructed with high accuracy (up to co-squareness). The proposed reconstruction is also interesting in itself from the point of view of graph theory.

In this paper, we introduce an adaptation of the counting sort algorithm that leverages the data obliviousness of the algorithm to enable the sorting of encrypted data using Fully Homomorphic Encryption (FHE). Our approach represents the first known sorting algorithm for encrypted data that does not rely on comparisons. The implementation takes advantage of some basic operations on TFHE's Look-Up-Tables (LUT). We have integrated these operations into RevoLUT, a comprehensive open-source library built on tfhe-rs. We demonstrate the effectiveness of our Blind Counting Sort algorithm by developing a top-k selection algorithm and applying it to privacy-preserving k-Nearest Neighbors classification. This proves to be approximately 4 times faster than state-of-the-art methods.

Graphs are a widely used data structure for collecting and analyzing relational data. However, when the graph structure is distributed across several parties, its analysis is challenging. In particular, due to the sensitivity of the data each party might want to keep their partial knowledge of the graph private, while still be willing to collaborate with the other parties for tasks of mutual benefit, such as data curation or the removal of poisoned data. To address this challenge, we propose Crypto'Graph, an efficient protocol for privacy-preserving link prediction on distributed graphs. More precisely, it allows parties partially sharing a graph with distributed links to infer the likelihood of formation of new links in the future. Through the use of cryptographic primitives, Crypto'Graph is able to compute the likelihood of these new links on the joint network without revealing the structure of the private graph of each party, even though they know the number of nodes they have, since they share the same graph in terms of nodes but not the same links. Crypto'Graph improves on previous works by enabling the computation of a diverse set of similarity metrics in parallel without any additional cost. The use of Crypto'Graph is illustrated for defense against graph poisoning attacks, in which potential adversarial links are identified without compromising the privacy of the graphs of individual parties. The effectiveness of Crypto'Graph in mitigating graph poisoning attacks and achieving high prediction accuracy on a node classification task using graph neural networks is demonstrated through extensive experimentation on two real-world datasets.