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A New Approach to Code Smoothing Bounds

Tsuyoshi Miezaki, Yusaku Nishimura, Katsuyuki Takashima

Abstract

To analyze the security of code-based cryptosystems, the smoothing parameter, which is closely related to the total variation distance of codes, has been investigated. While previous studies have bounded this distance using the Fourier transform on locally compact abelian groups, we take an alternative approach based on random walks. In this paper, we derive an inequality for the total variation distance of random walks using equitable partitions, and we show that our proposed bound generalizes existing results for finite abelian groups.

A New Approach to Code Smoothing Bounds

Abstract

To analyze the security of code-based cryptosystems, the smoothing parameter, which is closely related to the total variation distance of codes, has been investigated. While previous studies have bounded this distance using the Fourier transform on locally compact abelian groups, we take an alternative approach based on random walks. In this paper, we derive an inequality for the total variation distance of random walks using equitable partitions, and we show that our proposed bound generalizes existing results for finite abelian groups.
Paper Structure (11 sections, 7 theorems, 42 equations)

This paper contains 11 sections, 7 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Our Results
  4. Outline
  5. Preliminaries
  6. Total variation distance of a code
  7. Random walk
  8. Previous work
  9. The upper bound on the total variation distance using equitable partitions
  10. The upper bound on the total variation distance of a random walk
  11. The upper bound on the total variation distance of a code

Key Result

Theorem 2.1

Theorems & Definitions (25)

  • Definition 1: Code, dual code
  • Definition 2: Convolution
  • Definition 3: Total variation distance
  • Definition 4
  • Definition 5: Random walk
  • Definition 6: Total variation distance of a random walk
  • Remark 1
  • Definition 7: Fourier transform
  • Theorem 2.1: DDRT2023
  • Definition 8: Equitable partition
  • ...and 15 more