Locally computable UOWHF with linear shrinkage

We study the problem of constructing locally computable Universal One-Way Hash Functions (UOWHFs) ℋ:{0, 1}ⁿ → {0, 1}^m. A construction with constant output locality, where every bit of the output depends only on a constant number of bits of the input, was established by [Applebaum, Ishai, and Kushilevitz, SICOMP 2006]. However, this construction suffers from two limitations: (1) It can only achieve a sub-linear shrinkage of n - m = n^{1 - ε} ; and (2) It has a super-constant input locality, i.e., some inputs influence a large super-constant number of outputs. This leaves open the question of realizing UOWHFs with constant output locality and linear shrinkage of n - m = εn, or UOWHFs with constant input locality and minimal shrinkage of n - m = 1. We settle both questions simultaneously by providing the first construction of UOWHFs with linear shrinkage, constant input locality, and constant output locality. Our construction is based on the one-wayness of "random" local functions - a variant of an assumption made by Goldreich (ECCC 2000). Using a transformation of [Ishai, Kushilevitz, Ostrovsky and Sahai, STOC 2008], our UOWHFs give rise to a digital signature scheme with a minimal additive complexity overhead: signing n-bit messages with security parameter κ takes only O(n + κ) time instead of O(nκ) as in typical constructions. Previously, such signatures were only known to exist under an exponential hardness assumption. As an additional contribution, we obtain new locally-computable hardness amplification procedures for UOWHFs that preserve linear shrinkage.