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Cryptographic Applications of Twisted Goppa Codes

Harshdeep Singh, Anuj Kumar Bhagat, Ritumoni Sarma, Indivar Gupta

TL;DR

The paper defines multi-twisted Goppa (MTG) codes as subfield subcodes of duals of generalized multi-twisted Reed-Solomon codes and proves a distance bound $d \ge t+1$ under suitable conditions. It develops an explicit Extended Euclidean Algorithm–based decoding for MTG codes with twists at arbitrary positions, enabling correction of up to $\left\lfloor t/2 \right\rfloor$ errors, and integrates MTG codes into the Niederreiter cryptosystem with security against partial key-recovery. It also introduces quasi-cyclic MTG constructions to reduce public-key size and provides parameter guidance ensuring post-quantum resilience. The work bridging Goppa-type and RS-type code families yields practical cryptographic schemes with strong structural resistance and scalable key compression strategies, along with a framework for analyzing related attacks and future extensions.

Abstract

This article defines multi-twisted Goppa (MTG) codes as subfield subcodes of duals of multi-twisted Reed-Solomon (MTRS) codes and examines their properties. We show that if $t$ is the degree of the MTG polynomial defining an MTG code, its minimum distance is at least $t + 1$ under certain conditions. Extending earlier methods limited to single twist at last position, we use the extended Euclidean algorithm to efficiently decode MTG codes with a single twist at any position, correcting up to $\left\lfloor \tfrac{t}{2} \right\rfloor$ errors. This decoding method highlights the practical potential of these codes within the Niederreiter public key cryptosystem (PKC). Furthermore, we establish that the Niederreiter PKC based on MTG codes is secure against partial key recovery attacks. Additionally, we also reduce the public key size by constructing quasi-cyclic MTG codes using a non-trivial automorphism group.

Cryptographic Applications of Twisted Goppa Codes

TL;DR

The paper defines multi-twisted Goppa (MTG) codes as subfield subcodes of duals of generalized multi-twisted Reed-Solomon codes and proves a distance bound under suitable conditions. It develops an explicit Extended Euclidean Algorithm–based decoding for MTG codes with twists at arbitrary positions, enabling correction of up to errors, and integrates MTG codes into the Niederreiter cryptosystem with security against partial key-recovery. It also introduces quasi-cyclic MTG constructions to reduce public-key size and provides parameter guidance ensuring post-quantum resilience. The work bridging Goppa-type and RS-type code families yields practical cryptographic schemes with strong structural resistance and scalable key compression strategies, along with a framework for analyzing related attacks and future extensions.

Abstract

This article defines multi-twisted Goppa (MTG) codes as subfield subcodes of duals of multi-twisted Reed-Solomon (MTRS) codes and examines their properties. We show that if is the degree of the MTG polynomial defining an MTG code, its minimum distance is at least under certain conditions. Extending earlier methods limited to single twist at last position, we use the extended Euclidean algorithm to efficiently decode MTG codes with a single twist at any position, correcting up to errors. This decoding method highlights the practical potential of these codes within the Niederreiter public key cryptosystem (PKC). Furthermore, we establish that the Niederreiter PKC based on MTG codes is secure against partial key recovery attacks. Additionally, we also reduce the public key size by constructing quasi-cyclic MTG codes using a non-trivial automorphism group.
Paper Structure (9 sections, 23 theorems, 66 equations, 4 algorithms)

This paper contains 9 sections, 23 theorems, 66 equations, 4 algorithms.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Twisted Goppa codes and their Decoding
  4. Decoding Algorithm for Twisted Goppa codes
  5. Complexity Analysis of Decoding
  6. Cryptosystem based on Twisted Goppa codes
  7. Strengthening Niederreiter PKC Against Partial Key-Recovery Attacks
  8. Quasi-Cyclic Twisted Goppa code
  9. CONCLUSIONS

Key Result

Theorem 1

h-k-2023 Let $\alpha_1, \alpha_2, \dots, \alpha_n \in \mathbb{F}_q$ be distinct and $k<n<q$ and $\bm{\mathbbm{t}}=(t_1,t_2,\dots,t_\ell)$, $\bm{\mathbbm{h}}=(h_1,h_2,\dots,h_\ell)$ and $\bm \eta=(\eta_1,\eta_2,\dots, \eta_\ell)$ be as per Definition def::mtrs. Then the multi-twisted RS code $\mathca is non-singular. Here, $A_\mathcal{I}$ and $B_\mathcal{I}$ are lower and upper-triangular $t_\ell\t

Theorems & Definitions (61)

  • Definition 1: Multi-twisted Reed Solomon code
  • Theorem 1
  • Definition 2: Multi-twisted Goppa codes
  • Proposition 1
  • proof
  • Example 1
  • Remark 1
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 51 more