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Smoothing Linear Codes by Rényi Divergence and Applications to Security Reduction

Hao Yan, Cong Ling

TL;DR

This work studies smoothing parameters for code-based cryptography through Rényi divergence, deriving optimal smoothing bounds for random linear codes for all $\alpha>1$ and extending the analysis to structured code families. By developing averaging lemmas and entropy-based analyses, the authors prove vanishing $D_{\alpha}$ under appropriate rate conditions and adapt the results to random self-dual and quasi-cyclic codes. As an application, they obtain an average-case to average-case reduction from the Learning Parity With Noise problem to average-case decoding using Bernoulli noise, framed within the same parameter regime as prior work but achieved via Rényi divergence. The results strengthen security reductions in code-based cryptography and provide a flexible smoothing framework aligned with channel resolvability perspectives, now applicable to a broader set of code families.

Abstract

The concept of the smoothing parameter plays a crucial role in both lattice-based and code-based cryptography, primarily due to its effectiveness in achieving nearly uniform distributions through the addition of noise. Recent research by Pathegama and Barg has determined the optimal smoothing bound for random codes under Rényi Divergence for any order $α\in (1, \infty)$ \cite{pathegama2024r}. Considering the inherent complexity of encoding/decoding algorithms in random codes, our research introduces enhanced structural elements into these coding schemes. Specifically, this paper presents a novel derivation of the smoothing bound for random linear codes, maintaining the same order of Rényi Divergence and achieving optimality for any $α\in (1,\infty)$. We extend this framework under KL Divergence by transitioning from random linear codes to random self-dual codes, and subsequently to random quasi-cyclic codes, incorporating progressively more structures. As an application, we derive an average-case to average-case reduction from the Learning Parity with Noise (LPN) problem to the average-case decoding problem. This reduction aligns with the parameter regime in \cite{debris2022worst}, but uniquely employs Rényi divergence and directly considers Bernoulli noise, instead of combining ball noise and Bernoulli noise.

Smoothing Linear Codes by Rényi Divergence and Applications to Security Reduction

TL;DR

This work studies smoothing parameters for code-based cryptography through Rényi divergence, deriving optimal smoothing bounds for random linear codes for all and extending the analysis to structured code families. By developing averaging lemmas and entropy-based analyses, the authors prove vanishing under appropriate rate conditions and adapt the results to random self-dual and quasi-cyclic codes. As an application, they obtain an average-case to average-case reduction from the Learning Parity With Noise problem to average-case decoding using Bernoulli noise, framed within the same parameter regime as prior work but achieved via Rényi divergence. The results strengthen security reductions in code-based cryptography and provide a flexible smoothing framework aligned with channel resolvability perspectives, now applicable to a broader set of code families.

Abstract

The concept of the smoothing parameter plays a crucial role in both lattice-based and code-based cryptography, primarily due to its effectiveness in achieving nearly uniform distributions through the addition of noise. Recent research by Pathegama and Barg has determined the optimal smoothing bound for random codes under Rényi Divergence for any order \cite{pathegama2024r}. Considering the inherent complexity of encoding/decoding algorithms in random codes, our research introduces enhanced structural elements into these coding schemes. Specifically, this paper presents a novel derivation of the smoothing bound for random linear codes, maintaining the same order of Rényi Divergence and achieving optimality for any . We extend this framework under KL Divergence by transitioning from random linear codes to random self-dual codes, and subsequently to random quasi-cyclic codes, incorporating progressively more structures. As an application, we derive an average-case to average-case reduction from the Learning Parity with Noise (LPN) problem to the average-case decoding problem. This reduction aligns with the parameter regime in \cite{debris2022worst}, but uniquely employs Rényi divergence and directly considers Bernoulli noise, instead of combining ball noise and Bernoulli noise.
Paper Structure (15 sections, 15 theorems, 78 equations)

This paper contains 15 sections, 15 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Rényi Entropy and Divergence
  4. Linear, Self-Dual, and Quasi-Cyclic Codes
  5. Smoothing Parameter And Learning Parity With Noise
  6. Inequalities
  7. Random Linear Code Smoothing
  8. Rényi Divergence for $\alpha\in (1,2)$
  9. Rényi Divergence for $\alpha\in \mathbb{N}$
  10. Exponents Analysis
  11. Rényi Divergence for $\alpha>2$
  12. Random Self-Dual Code Smoothing
  13. Random Quasi Cyclic Code Smoothing
  14. Average to Average Case Reduction
  15. Conclusion

Key Result

Lemma 1

Let $a_1 \leq a_2 \leq \dots \leq a_n$ and $b_1 \leq b_2 \leq \dots \leq b_n$ be two sequences of real numbers. Then their rearrangement yields the maximum value when the sequences are ordered in the same way, that is, for any permutation $\sigma$ of $\{1, 2, \dots, n\}$.

Theorems & Definitions (30)

  • Definition 1: Linear Code
  • Definition 2: Self-Dual Code
  • Definition 3
  • Definition 4: LPN problem
  • Definition 5
  • Lemma 1: Rearrangement Inequality
  • Lemma 2
  • proof
  • Lemma 3: Averaging Lemma for Linear Codes loeliger1994basic
  • Theorem 1
  • ...and 20 more