AI 中文总结
本文通过构造显式同构并证明其会扭曲样本，论证了在完全分裂设置下基于根的扩展攻击无法产生新的漏洞。
AI 中文摘要
在本文中，我们解决了作者近期工作中提出的基于根的攻击泛化的一些关键问题。具体而言，我们通过从脆弱实例构造显式同构来分析潜在的基于根的扩展攻击，并提供了形式化证明，表明该方法在完全分裂设置下不会产生任何新的漏洞。为此，我们首先构造了完全分裂多项式环与先前攻击适用的多项式环之间的显式同构，并证明应用该同构总会扭曲样本，使得结果样本无法用于区分。然后，我们证明了完全分裂多项式环之间的任何同构都必须具有所构造同构的形式。
英文摘要
In the present work we address some key questions regarding the generalization of root-based attacks presented in [2]. In particular, we analyze potential root-based attacks extensions via the construction of explicit isomorphisms from vulnerable instances, and provide a formal proof that this approach will not yield any new vulnerabilities. To do so, we first construct an explicit isomorphism between fully-split polynomial rings and polynomial rings where previous attacks apply and show that the application of such an isomorphism will always distort the samples in a way that the resulting samples cannot be used to distinguish. Then, we prove that any isomorphism between fully-split polynomial rings must be of the form of the constructed isomorphism. Lastly, we study the generalization beyond fully split settings and show that the same distortion happens, regardless of the degree of the field extension.
