Accurate, Secure, and Efficient Semi-Constrained Navigation Over Encrypted City Maps

Navigation services enable users to find the shortest path from a starting point <inline-formula><tex-math notation="LaTeX">$S$</tex-math></inline-formula> to a destination <inline-formula><tex-math notation="LaTeX">$D$</tex-math></inline-formula>, reducing time, gas, and traffic congestion. Still, navigation users risk the exposure of their sensitive location data. Our motivation arises from how users can accurately, securely, and efficiently navigate from <inline-formula><tex-math notation="LaTeX">$S$</tex-math></inline-formula> to <inline-formula><tex-math notation="LaTeX">$D$</tex-math></inline-formula> while passing through <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> <i>unordered stops</i>, i.e., midway locations with a non-fixed visiting order. In this work, we formally define Semi-Constrained Navigation (SCN) and present a novel scheme Hermes to achieve accurate, secure, and efficient SCN. Specifically, we propose a divide-and-conquer approach to strike a good balance between accuracy and efficiency. It recursively depth-first-searches the whole area (a navigation tree) and invokes five carefully-crafted strategies stop-by-stop to compute three subpaths in three sequential subareas. We construct a path-distance oracle to encrypt the road graph and securely implement the strategies by using homomorphic encryption and garble circuits. We formally prove the security in the random oracle model and analyze the search complexity to be less than <inline-formula><tex-math notation="LaTeX">$O(k^{2})$</tex-math></inline-formula>. We experiment over a real-world city map and compare with six baselines. Results show that path search with <inline-formula><tex-math notation="LaTeX">$k=4$</tex-math></inline-formula> among <inline-formula><tex-math notation="LaTeX">$N=1000$</tex-math></inline-formula> intersections requires 5.58 seconds with a 3.2% distance deviation rate and an 82.5% path similarity.