Paper 2025/751

Improved Range Searching And Range Emptiness Under FHE Using Copy-And-Recurse

Abstract

Range counting is the problem of preprocessing a set $P\subset R^d$ of $n$ points, such that given a query range $\gamma$ we can efficiently compute $|P\cap\gamma|$. In the more general range searching problem the goal is to compute $f(P\cap\gamma)$, for some function $f$. It was already shown (Kushnir et al. PETS'24) how to efficiently answer a range searching query under FHE using a technique they called Copy-and-Recurse to traverse partition trees. In the Range emptiness problem the goal is to compute only whether $P\cap\gamma =\emptyset$. This was shown (in plaintext) to be done more efficiently. Range emptiness is interesting on its own and also used as a building block in other algorithms. In this paper we improve and extend the results of Kushnir et al. First, for range searching we reduce the overhead term to the optimal $O(n)$, so for example if the ranges are halfspaces in $R^d$ bounded by hyperplanes then range searching can be done with a circuit of size $O(t\cdot n^{1-1/d+\varepsilon}+n)$, where $t$ is the size of the sub-circuit that checks whether a point lies under a hyperplane. Second, we introduce a variation of copy-and-recurse that we call leveled copy-and-recurse. With this variation we improve range searching in the 1-dimensional case as well as traversal of other trees (e.g., binary trees and B-trees). Third, we show how to answer range emptiness queries under FHE more efficiently than range counting. We implemented our algorithms and show that our techniques for range emptiness yield a solution that is $\times 3.6$ faster than the previous results for a database of $2^{25}$ points.