$K$ -hop Reachability Query (KRQ) is one of fundamental graph queries, which can answer whether a node $u$ can reach a node $v$ within $K$ hops. With the scale of graph data increasing, data owner desires to outsource the local graphs to cloud server. To protect the graph privacy, data owner encrypts graphs before outsourcing them to the cloud server. It imposes a great challenge to KRQ over encrypted graphs. How to realize Privacy-Preserving $K$ -hop Reachability Query (PPKRQ) over encrypted graphs is still an unexplored problem. In this article, we explore this problem and propose a practical scheme. In order to efficiently answer KRQ over encrypted graphs, we construct the encrypted Breadth-First Spanning Tree table and adjacent list D (BFST-D). Based on encrypted BFST table, we can directly judge whether two query nodes are reachable within $K$ hops when they are in one spanning tree. The encrypted adjacent list D can help answer that two query nodes in different spanning trees. To protect the privacy, we utilize the Paillier cryptographic and Order-Revealing Encryption (ORE) to support the comparison and computation over ciphertexts. As a result, our scheme achieves the sensitive information privacy without losing the ability of querying over encrypted graphs. The security analysis shows that our proposed scheme is secure based on semi-honest cloud server. The extensive experiments show the efficiency of our scheme.

中文翻译：

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在加密图上启用隐私保护 $K$K-Hop 可达性查询


$K$ 跳可达性查询（KRQ）是基本图查询之一，它可以回答节点 $u$ 是否可以在 $K$ 跳内到达节点 $v$。随着图数据规模的不断增大，数据所有者希望将本地图外包给云服务器。为了保护图隐私，数据所有者在将图外包到云服务器之前对图进行加密。这对加密图上的KRQ提出了巨大的挑战。如何在加密图上实现隐私保护$K$跳可达性查询（PPKRQ）仍然是一个尚未探索的问题。在本文中，我们探讨了这个问题并提出了一个实用的方案。为了有效地回答加密图上的KRQ，我们构造了加密的广度优先生成树表和相邻列表D（BFST-D）。基于加密的BFST表，我们可以直接判断两个查询节点在一棵生成树中时是否在$K$跳内可达。加密的相邻列表D可以帮助回答不同生成树中的两个查询节点。为了保护隐私，我们利用 Paillier 密码学和 Order-Revealing Encryption (ORE) 来支持密文的比较和计算。因此，我们的方案在不失去加密图查询能力的情况下实现了敏感信息隐私。安全分析表明，我们提出的基于半诚实云服务器的方案是安全的。大量的实验表明了我们方案的有效性。