Improved Search-to-Decision Reduction for Random Local Functions
Kel Zin Tan, Prashant Nalini Vasudevan
TL;DR
The paper addresses the hardness of inverting random local functions defined by a constant-arity predicate, a cornerstone problem in local cryptography and CSP-inspired constructions. It introduces a novel search-to-decision reduction that works for any constant-arity predicate, removing previous sensitivity requirements and achieving inversion with outputs on the order of $\tilde{O}\big(m(n/\varepsilon)^2\big)$ and success $\Omega(\varepsilon)$, while also extending to polylogarithmic arity and noisy predicates. The approach hinges on a randomised hypergraph transformation that mixes instances, coupled with a predictor built from a decision oracle and a hybrid argument to recover the secret input; amplification yields the full recovery of the input with polynomial runtime. This work broadens the potential for strong local PRGs and deepens the connections between decision hardness, CSP-like structures, and cryptographic primitives, offering techniques likely applicable beyond random local functions. Overall, the results demonstrate robust search-to-decision reductions without requiring structural properties of the predicate, advancing cryptographic constructions based on local, low-depth computations and their practical implications for PRGs and related primitives.
Abstract
A random local function defined by a $d$-ary predicate $P$ is one where each output bit is computed by applying $P$ to $d$ randomly chosen bits of its input. These represent natural distributions of instances for constraint satisfaction problems. They were put forward by Goldreich as candidates for low-complexity one-way functions, and have subsequently been widely studied also as potential pseudo-random generators. We present a new search-to-decision reduction for random local functions defined by any predicate of constant arity. Given any efficient algorithm that can distinguish, with advantage $ε$, the output of a random local function with $m$ outputs and $n$ inputs from random, our reduction produces an efficient algorithm that can invert such functions with $\tilde{O}(m(n/ε)^2)$ outputs, succeeding with probability $Ω(ε)$. This implies that if a family of local functions is one-way, then a related family with shorter output length is family of pseudo-random generators. Prior to our work, all such reductions that were known required the predicate to have additional sensitivity properties, whereas our reduction works for any predicate. Our results also generalise to some super-constant values of the arity $d$, and to noisy predicates.