Locality-Preserving Hashing for Shifts with Connections to Cryptography

Abstract

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 → Z n is a (d, δ) localitypreserving hash function for shifts (LPHS) if: (1) h can be computed by (adaptively) querying d bits of its input, and (2) Pr [h(x) = h(x ≪ 1) + 1] ≤ δ, where x is random and ≪ 1 denotes a cyclic shift by one bit to the left. We make the following contributions.