Klee’s Measure Problem Made Oblivious

Abstract

AbstractWe study Klee’s measure problem — computing the volume of the union of n axis-parallel hyperrectangles in $$\mathbb {R}^d$$ R d — in the oblivious RAM (ORAM) setting. For this, we modify Chan’s algorithm [12] to guarantee memory access patterns and control flow independent of the input; this makes the resulting algorithm applicable to privacy-preserving computation over outsourced data and (secure) multi-party computation.For $$d = 2$$ d = 2 , we develop an oblivious version of Chan’s algorithm that runs in expected $$\mathcal {O}(n \log ^{5/3} n)$$ O ( n log 5 / 3 n ) time for perfect security or $$\mathcal {O}(n \log ^{3/2} n)$$ O ( n log 3 / 2 n ) time for computational security, thus improving over optimal general transformations. For $$d \ge 3$$ d ≥ 3 , we obtain an oblivious version with perfect security while maintaining the $$\mathcal {O}(n^{d/2})$$ O ( n d / 2 ) runtime, i. e., without any overhead.Generalizing our approach, we derive a technique to transform divide-and-conquer algorithms that rely on linear-scan processing into oblivious counterparts. As such, our results are of independent interest for geometric divide-and-conquer algorithms that maintain an order over the input. We apply our technique to two such algorithms and obtain efficient oblivious counterparts of algorithms for inversion counting and computing a closest pair in two dimensions.