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On the Security and Design of Cryptosystems Using Gabidulin-Kronecker Product Codes

Terry Shue Chien Lau, Zhe Sun, Sook-Chin Yip, Ji-Jian Chin, Choo-Yee Ting

TL;DR

This paper is a preliminary study on the security and design of cryptosystems using Gabidulin-Kronecker Product Codes, and proposes ways to improve it.

Abstract

This paper is a preliminary study on the security and design of cryptosystems using Gabidulin-Kronecker Product Codes. In particular, we point out the design impracticality of the system, and propose ways to improve it.

On the Security and Design of Cryptosystems Using Gabidulin-Kronecker Product Codes

TL;DR

This paper is a preliminary study on the security and design of cryptosystems using Gabidulin-Kronecker Product Codes, and proposes ways to improve it.

Abstract

This paper is a preliminary study on the security and design of cryptosystems using Gabidulin-Kronecker Product Codes. In particular, we point out the design impracticality of the system, and propose ways to improve it.

Paper Structure

This paper contains 17 sections, 6 theorems, 24 equations, 1 table.

Table of Contents

  1. GabKron Cryptosystem
  2. Analysis on the Gabidulin-Kronecker Product Code
  3. Encoding and Decoding
  4. Subcode of Gabidulin-block code
  5. Analysis on the Design of GabKron Cryptosystem
  6. On the choice of $X$
  7. On the Left Scrambler Matrix $S$
  8. On the Key Sizes
  9. On the Parameter Sets
  10. Repaired Parameter Sets
  11. Improved GabKron Cryptosystem with Partial-Circulant-Block Structure
  12. General Idea for Improved GabKron Cryptosystem
  13. Modification on $G$
  14. Construction of $X$
  15. Construction of $P$
  16. ...and 2 more sections

Key Result

Lemma 1

Suppose that $G = G_1 \otimes G_2$ is a generator matrix for an $[n,k]$-Gabidulin-Kronecker product code defined as above. Then the matrix $\bar{G}_1 = \left[ \right] \in \mathbb{F}_{q^m}^{k_1k_2 \times n_1 k_2}$ has $rank(\bar{G}_1) = k$.

Theorems & Definitions (14)

  • Definition 1: SZZF2024
  • Lemma 1
  • proof
  • Definition 2
  • Corollary 2
  • proof
  • Remark 1
  • Proposition 3
  • proof
  • Definition 3
  • ...and 4 more