Cryptographic Applications of Twisted Goppa Codes

TL;DR

This paper introduces multi-twisted Goppa codes, explores their properties, efficient decoding methods, security in cryptographic applications, and reduces key sizes using automorphisms, enhancing their practical cryptographic utility.

Contribution

It defines multi-twisted Goppa codes, extends decoding techniques, analyzes security in Niederreiter cryptosystems, and constructs quasi-cyclic variants to reduce key size.

Findings

Minimum distance at least t+1 under certain conditions

Efficient decoding correcting up to ⌊t/2⌋ errors

Security against partial key recovery attacks

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.