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.

翻译：本文定义了多扭曲戈帕（MTG）码作为多扭曲里德-所罗门（MTRS）码对偶码的子域子码，并研究了其性质。我们证明，若$t$为定义MTG码的多扭曲戈帕多项式次数，在特定条件下其最小距离至少为$t + 1$。通过扩展先前局限于末位单扭曲的方法，我们利用扩展欧几里得算法实现了对任意位置单扭曲MTG码的高效解码，可纠正最多$\left\lfloor \tfrac{t}{2} \right\rfloor$个错误。该解码方法凸显了此类码在尼德赖特公钥密码系统（PKC）中的实际应用潜力。此外，我们证明了基于MTG码的尼德赖特PKC能够抵御部分密钥恢复攻击。另外，通过利用非平凡自同构群构造拟循环MTG码，我们还实现了公钥规模的缩减。