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Quasi-cyclic Linear Error-Block Code-based Post-quantum Signature

I. Cherkaoui, S. Belabssir, J. Horgan, I. Dey

TL;DR

A signature based on a family of linear error-block codes (LEB), with strong algebraic properties is introduced, being quantum-resistant and reaching the Gilbert-Varshamov bound, thus offering a good trade-off between rate and distance.

Abstract

Shor algorithm led to the discovery of multiple vulnerabilities in a number of cryptosystems. As a result, post-quantum cryptography attempts to provide cryptographic solutions that can face these attacks, ensuring the security of sensitive data in a future where quantum computers are assumed to exist. Error correcting codes are a source for efficiency when it comes to signatures, especially random ones described in this paper, being quantum-resistant and reaching the Gilbert-Varshamov bound, thus offering a good trade-off between rate and distance. In the light of this discussion, we introduce a signature based on a family of linear error-block codes (LEB), with strong algebraic properties: it is the family of quasi-cyclic LEB codes that we do define algebraically during this work.

Quasi-cyclic Linear Error-Block Code-based Post-quantum Signature

TL;DR

A signature based on a family of linear error-block codes (LEB), with strong algebraic properties is introduced, being quantum-resistant and reaching the Gilbert-Varshamov bound, thus offering a good trade-off between rate and distance.

Abstract

Shor algorithm led to the discovery of multiple vulnerabilities in a number of cryptosystems. As a result, post-quantum cryptography attempts to provide cryptographic solutions that can face these attacks, ensuring the security of sensitive data in a future where quantum computers are assumed to exist. Error correcting codes are a source for efficiency when it comes to signatures, especially random ones described in this paper, being quantum-resistant and reaching the Gilbert-Varshamov bound, thus offering a good trade-off between rate and distance. In the light of this discussion, we introduce a signature based on a family of linear error-block codes (LEB), with strong algebraic properties: it is the family of quasi-cyclic LEB codes that we do define algebraically during this work.

Paper Structure

This paper contains 15 sections, 9 theorems, 66 equations, 15 figures, 6 tables, 9 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. LEB cyclic codes
  4. Algebraic representation
  5. Polynomial Representation
  6. Quasi-cyclic LEB codes
  7. Definition and Properties
  8. Polynomial Representation
  9. Decoding quasi-cyclic LEB Codes
  10. NP-completeness of decoding QC LEB codes
  11. Meggitt-Like Decoding of QC-LEB Codes
  12. Complexity
  13. The code based signature
  14. Security and performance analysis
  15. Conclusion

Key Result

Proposition 3.5

Let $\pi$ be a partition of a positive integer $n$ of the form $\pi = [m]^s$. Let $\mathbb{F}_q$ be the finite field of $q$ elements and $R_{\pi}=\frac{\mathbb{F}_q[X]}{<X^n-1>}$ be the quotient ring. Then $(R_{\pi},+,\star)$ where $+$ is the classical addition of polynomial and $\star$ is the polyn

Figures (15)

  • Figure 1: Input time and size for the signature
  • Figure 2: Auto-correlation of the sequence bits
  • Figure 3: NIST Security of QC-LEB according to key and signature size
  • Figure 4: Graph comparing BER vs SNR for QC-LEB against Distributed Turbo and LDPC
  • Figure 5: QC-LEB code performance comparative analysis
  • ...and 10 more figures

Theorems & Definitions (37)

  • Definition 3.1
  • Remark 3.2
  • Example 3.3
  • Definition 3.4: The binary operation $\star$
  • Proposition 3.5
  • proof
  • Theorem 3.6
  • proof
  • Definition 3.7
  • Definition 3.8
  • ...and 27 more