Improved Search-to-Decision Reduction for Random Local Functions

2025-10-03

AI Summary

This work investigates efficient reductions from search to decision problems for Random Local Functions (RLFs). For RLFs defined by arbitrary constant-arity predicates, we construct the first generic search-to-decision reduction framework that does not rely on sensitivity assumptions, extending it to certain super-constant arities and noisy settings. Methodologically, we combine multi-output extension, error amplification, and a distinguisher-to-inverter conversion technique to efficiently transform an ε-advantage distinguisher into an inverter with Ω(ε) success probability, using only Õ(m(n/ε)²) outputs. Our results strengthen the theoretical connection between one-wayness and pseudorandomness: if a class of RLFs is one-way, then—by appropriately shortening the output length—one can still derive a secure pseudorandom generator.

Technology Category

• Search and Optimization: Local Search
• Machine Learning: Adversarial Learning & Robustness
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Applications of cryptography
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

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 $ ilde{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.

Problem

• Develops search-to-decision reduction for random local functions
• Inverts functions using distinguishing advantage with polynomial outputs
• Works for arbitrary constant-arity predicates without extra properties

Innovation

• Search-to-decision reduction for random local functions
• Works for any constant-arity predicate without restrictions
• Generalizes to super-constant arity and noisy predicates