Abstract

Starting from the problem of d-tensor isomorphism (d- \(\textsf {TI}\) ), we study the relation between various code equivalence problems in different metrics. In particular, we show a reduction from the sum-rank metric ( \(\textsf {CE}_{\textsf {sr}}\) ) to the rank metric ( \(\textsf {CE}_{\textsf {rk}}\) ). To obtain this result, we investigate reductions between tensor problems. We define the monomial isomorphism problem for d-tensors (d- \(\textsf {TI}^*\) ), where, given two d-tensors, we ask if there are \(d-1\) invertible matrices and a monomial matrix sending one tensor into the other. We link this problem to the well-studied d- \(\textsf {TI}\) and the \(\textsf {TI}\) -completeness of d- \(\textsf {TI}^*\) is shown. Due to this result, we obtain a reduction from \(\textsf {CE}_{\textsf {sr}}\) to \(\textsf {CE}_{\textsf {rk}}\) . In the literature, a similar result was known, but it needs an additional assumption on the automorphisms of matrix codes. Since many constructions based on the hardness of Code Equivalence problems are emerging in cryptography, we analyze how such reductions can be taken into account in the design of cryptosystems based on \(\textsf {CE}_{\textsf {sr}}\) .

中文翻译：

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张量的单项同构及其在代码等价问题中的应用

摘要

从d张量同构问题（d - \(\textsf {TI}\)）出发，研究不同度量下各种代码等价问题之间的关系。特别是，我们展示了从总和度量（\(\textsf {CE}_{\textsf {sr}}\) ）到排序度量（\(\textsf {CE}_{\textsf {rk }}\) )。为了获得这个结果，我们研究了张量问题之间的约简。我们定义d张量 ( d - \(\textsf {TI}^*\) )的单项式同构问题，其中，给定两个d张量，我们询问是否存在\(d-1\)个可逆矩阵和一个单项式矩阵将一个张量发送到另一个张量。我们将此问题与经过充分研究的d - \(\textsf {TI}\)联系起来，并显示了d - \( \textsf { TI}^*\)的 \(\textsf {TI}\) 完整性。由于这个结果，我们得到了从\(\textsf {CE}_{\textsf {sr}}\)到\(\textsf {CE}_{\textsf {rk}}\) 的减少。在文献中，已知有类似的结果，但需要对矩阵码的自同构进行额外的假设。由于密码学中出现了许多基于代码等价问题的难度的构造，我们分析了在基于\(\textsf {CE}_{\textsf {sr}}\)的密码系统设计中如何考虑这种减少。