Cryptographic Applications of Twisted Goppa Codes

2026-02-18

AI Summary

This work addresses the challenges of security and efficiency in code-based public-key cryptosystems by proposing Multi-Twisted Goppa (MTG) codes—defined as subfield subcodes of the duals of multi-twisted Reed–Solomon codes—and integrating them into the Niederreiter framework. An efficient decoding algorithm, based on the extended Euclidean algorithm and supporting single twists at arbitrary positions, enables reliable correction of up to ⌊t/2⌋ errors. Furthermore, a nontrivial automorphism group is leveraged to construct a quasi-cyclic structure, substantially reducing public key size. Theoretical analysis demonstrates that the proposed scheme effectively resists certain key-recovery attacks while significantly lowering storage overhead without compromising security.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Multi-instance/Multi-view Learning
• Multiagent Systems: Mechanism Design

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Data and user privacy-enhancing technologies for the Web

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.

Problem

• multi-twisted Goppa codes
• Niederreiter public key cryptosystem
• partial key recovery attacks
• public key size
• code-based cryptography

Innovation

• multi-twisted Goppa codes
• efficient decoding
• Niederreiter cryptosystem
• partial key recovery resistance
• quasi-cyclic construction