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Leakage-Resilient Hardness Equivalence to Logspace Derandomization

Yakov Shalunov

TL;DR

The paper establishes a space-bounded counterpart to Liu and Pass's hardness–derandomization equivalence by introducing leakage-resilient average hardness in logspace and proving an equivalence with derandomizing $prBP{\boldsymbol{\cdot}}L$ via one-sided search. It shows how a leakage-resilient hard function yields a logspace pseudorandom generator and, consequently, derandomizes logspace promise-BP computations, while also proving the converse: logspace derandomization implies a leakage-resilient hard function under an average-case framework. The key innovations include a logspace-adapted NW generator with a leakage-based hardness replacement, a one-sided search reduction for promise problems, and a strengthened average-case hardness notion to accommodate logspace limitations. The results bridge hardness assumptions and logspace derandomization, advancing conditional derandomization theory in space-bounded settings and offering a foundation for further reductions and potential unconditional progress.

Abstract

Efficient derandomization has long been a goal in complexity theory, and a major recent result by Yanyi Liu and Rafael Pass identifies a new class of hardness assumption under which it is possible to perform time-bounded derandomization efficiently: that of ''leakage-resilient hardness.'' They identify a specific form of this assumption which is $\textit{equivalent}$ to $\mathsf{prP} = \mathsf{prBPP}$. In this paper, we pursue an equivalence to derandomization of $\mathsf{prBP{\cdot}L}$ (logspace promise problems with two-way randomness) through techniques analogous to Liu and Pass. We are able to obtain an equivalence between a similar ''leakage-resilient hardness'' assumption and a slightly stronger statement than derandomization of $\mathsf{prBP{\cdot}L}$, that of finding ''non-no'' instances of ''promise search problems.''

Leakage-Resilient Hardness Equivalence to Logspace Derandomization

TL;DR

The paper establishes a space-bounded counterpart to Liu and Pass's hardness–derandomization equivalence by introducing leakage-resilient average hardness in logspace and proving an equivalence with derandomizing via one-sided search. It shows how a leakage-resilient hard function yields a logspace pseudorandom generator and, consequently, derandomizes logspace promise-BP computations, while also proving the converse: logspace derandomization implies a leakage-resilient hard function under an average-case framework. The key innovations include a logspace-adapted NW generator with a leakage-based hardness replacement, a one-sided search reduction for promise problems, and a strengthened average-case hardness notion to accommodate logspace limitations. The results bridge hardness assumptions and logspace derandomization, advancing conditional derandomization theory in space-bounded settings and offering a foundation for further reductions and potential unconditional progress.

Abstract

Efficient derandomization has long been a goal in complexity theory, and a major recent result by Yanyi Liu and Rafael Pass identifies a new class of hardness assumption under which it is possible to perform time-bounded derandomization efficiently: that of ''leakage-resilient hardness.'' They identify a specific form of this assumption which is to . In this paper, we pursue an equivalence to derandomization of (logspace promise problems with two-way randomness) through techniques analogous to Liu and Pass. We are able to obtain an equivalence between a similar ''leakage-resilient hardness'' assumption and a slightly stronger statement than derandomization of , that of finding ''non-no'' instances of ''promise search problems.''
Paper Structure (15 sections, 11 theorems, 4 equations)

This paper contains 15 sections, 11 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Informal description of Liu and Pass's polynomial-time equivalence result
  4. Forward direction (Liu and Pass)
  5. Backward direction (Liu and Pass)
  6. Results
  7. Preliminaries
  8. Existence of hard function implies derandomization
  9. Hard function implies existence of PRG
  10. Derandomization of search problems using the PRG
  11. Derandomization implies existence of hard function
  12. Random is hard
  13. Construction of hard function
  14. Discussion
  15. Proof of lemmas for derandomization using PRG

Key Result

Theorem 1

There is a logspace-computable function which is leakage-resilient hard if and only if it is possible, in logspace, to deterministically solve a one-sided search for any $\mathsf{prBP{\boldsymbol{\cdot}}L}$ search problem.

Theorems & Definitions (18)

  • Theorem 1: Main Theorem (informal)
  • Definition 2: Almost-all input leakage-resilient hardness liu_2022rivest_1985chen_2021
  • Theorem 3: Liu and Pass liu_2022
  • Definition 4: $\prBPP$ search problems goldreich_2011
  • Theorem 5: Main theorem
  • Definition 6: $\mathsf{prBP{\boldsymbol{\cdot}}L}$ search problems
  • Definition 7: Almost-all-input leakage resilient average hard
  • Theorem 8: Existence implies derandomization
  • Definition 9: Targeted distinguisher
  • Definition 10: Uniform targeted pseudorandom generator
  • ...and 8 more