Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Projective Systematic Authentication via Reed-Muller Codes

Hsuan-Po Liu, Hessam Mahdavifar

TL;DR

The paper addresses authenticating messages without secrecy by building projective systematic authentication schemes from binary error-correcting codes, focusing on Reed-Muller codes. It introduces RM-A-codes, a two-part key and projection scheme that maps a source to a higher-dimensional RM codeword, projects to a lower-dimension, and masks to generate a tag, with a last-entry constraint on the codeword to control deception. The authors derive closed-form expressions for the impersonation probability $P_\mathrm{I}$ and provide a tractable, RM-structure-based formula for the substitution probability $P_\mathrm{S}$, highlighting how RM-code properties influence security. Numerical results show stable $P_\mathrm{I}$ and favorable reductions in $P_\mathrm{S}$ with increasing blocklength, suggesting practical, low-complexity authentication suitable for IoT settings.

Abstract

In this paper, we study the problem of constructing projective systematic authentication schemes based on binary linear codes. In systematic authentication, a tag for authentication is generated and then appended to the information, also referred to as the source, to be sent from the sender. Existing approaches to leverage projective constructions focus primarily on codes over large alphabets, and the projection is simply into one single symbol of the codeword. In this work, we extend the projective construction and propose a general projection process in which the source, which is mapped to a higher dimensional codeword in a given code, is first projected to a lower dimensional vector. The resulting vector is then masked to generate the tag. To showcase the new method, we focus on leveraging binary linear codes and, in particular, Reed-Muller (RM) codes for the proposed projective construction. More specifically, we propose systematic authentication schemes based on RM codes, referred to as RM-Acodes. We provide analytical results for probabilities of deception, widely considered as the main metrics to evaluate the performance of authentication systems. Through our analysis, we discover and discuss explicit connections between the probabilities of deception and various properties of RM codes.

Projective Systematic Authentication via Reed-Muller Codes

TL;DR

The paper addresses authenticating messages without secrecy by building projective systematic authentication schemes from binary error-correcting codes, focusing on Reed-Muller codes. It introduces RM-A-codes, a two-part key and projection scheme that maps a source to a higher-dimensional RM codeword, projects to a lower-dimension, and masks to generate a tag, with a last-entry constraint on the codeword to control deception. The authors derive closed-form expressions for the impersonation probability and provide a tractable, RM-structure-based formula for the substitution probability , highlighting how RM-code properties influence security. Numerical results show stable and favorable reductions in with increasing blocklength, suggesting practical, low-complexity authentication suitable for IoT settings.

Abstract

In this paper, we study the problem of constructing projective systematic authentication schemes based on binary linear codes. In systematic authentication, a tag for authentication is generated and then appended to the information, also referred to as the source, to be sent from the sender. Existing approaches to leverage projective constructions focus primarily on codes over large alphabets, and the projection is simply into one single symbol of the codeword. In this work, we extend the projective construction and propose a general projection process in which the source, which is mapped to a higher dimensional codeword in a given code, is first projected to a lower dimensional vector. The resulting vector is then masked to generate the tag. To showcase the new method, we focus on leveraging binary linear codes and, in particular, Reed-Muller (RM) codes for the proposed projective construction. More specifically, we propose systematic authentication schemes based on RM codes, referred to as RM-Acodes. We provide analytical results for probabilities of deception, widely considered as the main metrics to evaluate the performance of authentication systems. Through our analysis, we discover and discuss explicit connections between the probabilities of deception and various properties of RM codes.
Paper Structure (12 sections, 12 theorems, 30 equations, 2 figures, 2 tables)

This paper contains 12 sections, 12 theorems, 30 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Systematic authentication codes
  4. Probability of deception
  5. RM codes: A brief overview
  6. The Proposed RM-A-codes
  7. Analysis
  8. The probability of success of the impersonation attack $P_\mathrm{I}$
  9. Analysis for $P_{\mathrm{S}}$: The probability of success of the substitution attack
  10. Numerical Analysis
  11. Conclusion
  12. Linearity of RM-A-codes

Key Result

Lemma 1

Given a generator matrix $\mathbf{G}$ from $\mathrm{RM}(m,r)$, we have $\mathbf{c}_\mathbf{s}+\mathbf{c}_{\mathbf{s}^\prime}=\mathbf{1}_n$ when $\mathbf{u=s}$ and $\mathbf{u}^\prime=\mathbf{s}^\prime$, where $\mathbf{c}_\mathbf{s}=\mathbf{uG}$ and $\mathbf{c}_{\mathbf{s}^\prime}=\mathbf{u}^\prime\ma

Figures (2)

  • Figure 1: Systematic authentication system
  • Figure 2: Two types of attacks

Theorems & Definitions (29)

  • Definition 1: RM-A-codes
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 3
  • proof
  • Example 1
  • Theorem 4
  • ...and 19 more