Abstract

In this paper, we analyze the hardness of the Matrix Code Equivalence (MCE) problem for matrix codes endowed with the rank metric, and provide the first algorithms for solving it. We do this by making a connection to another well-known equivalence problem from multivariate cryptography—the Isomorphism of Polynomials (IP). Under mild assumptions, we give tight reductions from MCE to the homogenous version of the Quadratic Maps Linear Equivalence (QMLE) problem, and vice versa. Furthermore, we present reductions to and from similar problems in the sum-rank metric, showing that MCE is at the core of code equivalence problems. On the practical side, using birthday techniques known for IP, we present two algorithms: a probabilistic algorithm for MCE running in time \(q^{\frac{2}{3}(n+m)}\) up to a polynomial factor, and a deterministic algorithm for MCE with roots, running in time \(q^{\min \{m,n,k\}}\) up to a polynomial factor. Lastly, to confirm these findings, we solve randomly-generated instances of MCE using these two algorithms.

中文翻译：

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排名度量中代码等价问题的硬度估计

摘要

在本文中，我们分析了赋予秩度量的矩阵码的矩阵码等价（ MCE ）问题的难度，并提供了第一个解决该问题的算法。我们通过与多元密码学中另一个众所周知的等价问题——多项式同构（ IP ） ——建立联系来做到这一点。在温和的假设下，我们将MCE严格简化为二次映射线性等价 ( QMLE ) 问题的齐次版本，反之亦然。此外，我们在总和度量中提出了类似问题的减少，表明MCE是代码等价问题的核心。在实际方面，使用IP已知的生日技术，我们提出了两种算法： MCE的概率算法，运行时间\(q^{\frac{2}{3}(n+m)}\)高达多项式因子，以及具有根的MC​​E确定性算法，运行时间\(q^{\min \{m,n,k\}}\)直至多项式因子。最后，为了证实这些发现，我们使用这两种算法求解随机生成的MCE实例。