发表机构
King’s College London; SandboxAQ; Royal Holloway, University of London; Aalto University（伦敦国王学院; SandboxAQ公司; 伦敦大学皇家霍洛威学院; 阿尔托大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出在奇特征有限域上均匀采样任意壳维数线性码的算法，首次实现任意码率自正交码采样，并研究完全正交性以支持PCE密码方案。
AI 中文摘要
我们给出了一种算法，可在奇特征的有限域上均匀随机采样具有任意壳维数和类型的线性码，这首次实现了对任意码率的均匀随机自正交码（包括自对偶码）的采样。我们的算法是对Albrecht、Benčina和Lai（EC'25）所给算法的重新审视，利用了Li、Shi和Ling（IEEE Trans. Inf. Theory 71(1)）证明的质量公式。这使得我们能够实例化基于码的密码方案，这些方案的安全性依赖于'随机'自对偶码上的置换码等价（PCE）问题的困难性，而此前无法对其进行采样。基于Bardet、Otmani和Saeed-Taha（ISIT'19）的观察——即在有限扩域上考虑PCE时，由于非平凡的伽罗瓦自同构，欧几里得正交性是不充分的——我们研究了所谓的完全正交性的行为，即同时相对于所有诱导的伽罗瓦几何的正交性。我们刻画了向量和码的完全正交性，并给出了一种算法，用于采样具有指定维数完全壳的线性码；该壳是环境空间所有伽罗瓦几何中充当壳的子码。该算法导致码率降低，降低因子等于扩域次数，然而，我们论证了在PCE的背景下这可能是必要的，并探讨了它如何限制我们算法的实用性。当伽罗瓦壳对称时，我们考虑了线性码的伽罗瓦类型的概念，并证明所有线性码都具有恒定的厄米特类型。
英文摘要
We give an algorithm that samples uniformly random linear codes of any hull dimension and type over finite fields of odd characteristic, implying the first algorithm for sampling uniformly random self-orthogonal codes of any rate, including self-dual codes. Our algorithm is a re-visitation of the algorithm given by Albrecht, Benčina and Lai (EC'25), using the mass formulae proven by Li, Shi and Ling (IEEE Trans. Inf. Theory 71(1)). This allows us to instantiate code-based cryptographic schemes that rely on the hardness of Permutation Code Equivalence (PCE) on `random' self-dual codes for security and that were previously unable to sample them. Building on the observation by Bardet, Otmani and Saeed-Taha (ISIT'19) that Euclidean orthogonality is insufficient when considering PCE over finite extension fields due to non-trivial Galois automorphisms, we study the behaviour of what we call total orthogonality, that is orthogonality with respect to all induced Galois geometries simultaneously. We characterise total orthogonality of vectors and codes, and give an algorithm that samples linear codes with a total hull of a prescribed dimension; a subcode that acts as the hull in all Galois geometries of the ambient space. The algorithm incurs a rate decrease by a factor equal to the extension degree, however, we argue why this may be necessary in the context of PCE and explore how it limits the practicality of our algorithm. We consider the notion of Galois type of a linear code when Galois hulls are symmetric and show that all linear codes have constant Hermitian type.
