Can we sense our location in an unfamiliar environment by taking a sublinear-size sample of our surroundings? Can we efficiently encrypt a message that only someone physically close to us can decrypt? To solve this kind of problems, we introduce and study a new type of hash functions for finding shifts in sublinear time. A function $h:\{0,1\}^n\to \mathbb{Z}_n$ is a $(d,\delta)$ {\em locality-preserving hash function for shifts} (LPHS) if: (1) $h$ can be computed by (adaptively) querying $d$ bits of its input, and (2) $\Pr [ h(x) \neq h(x \ll 1) + 1 ] \leq \delta$, where $x$ is random and $\ll 1$ denotes a cyclic shift by one bit to the left. We make the following contributions. * Near-optimal LPHS via Distributed Discrete Log: We establish a general two-way connection between LPHS and algorithms for distributed discrete logarithm in the generic group model. Using such an algorithm of Dinur et al. (Crypto 2018), we get LPHS with near-optimal error of $\delta=\tilde O(1/d^2)$. This gives an unusual example for the usefulness of group-based cryptography in a post-quantum world. We extend the positive result to non-cyclic and worst-case variants of LPHS. * Multidimensional LPHS: We obtain positive and negative results for a multidimensional extension of LPHS, making progress towards an optimal 2-dimensional LPHS. * Applications: We demonstrate the usefulness of LPHS by presenting cryptographic and algorithmic applications. In particular, we apply multidimensional LPHS to obtain an efficient "packed" implementation of homomorphic secret sharing and a sublinear-time implementation of location-sensitive encryption whose decryption requires a significantly overlapping view.
翻译:我们能否在不熟悉的环境中通过对周围环境进行亚直线大小的多层面样本来感知我们的位置? 我们能否有效地加密一个只有实际接近我们的人才能解密的信息? 为了解决这种类型的问题, 我们引入并研究一种新的散列函数类型, 以寻找亚直线时间的转变。 一个函数 $: 0. 1\\\\\\\\\\ mathb ⁇ 到\\\\\\\ mathb\\ 美元, 美元( d,\ delta) 美元 = 本地保存 hash 函数移转 (LPHS) : (1) 可以通过( 调整) 查询其输入的 $d) 来计算美元 。 (We-optimal LPHS 应用( 调整) 2 d) 来计算其输入的值 ; (2) $\ (x) lPHS (x) h=xxxx) 显示一个正向左移动的流转动 。 我们做出以下贡献 。 * 近 OPHS- sal- dal- dalmagial ladeal ladeal ladeal laft laft ex ex ex ex ladeal ex ex ex ex laveal a grocudeal ex ex ex ex laut a grocudeal exm sal ex ex ex ex exmal ex ex exmal exmal 。