Skew polynomials are a class of non-commutative polynomials that have several applications in computer science, coding theory and cryptography. In particular, skew polynomials can be used to construct and decode evaluation codes in several metrics, like e.g. the Hamming, rank, sum-rank and skew metric. We propose a fast divide-and-conquer variant of Kötter–Nielsen–Høholdt (KNH) interpolation algorithm: it inputs a list of linear functionals on skew polynomial vectors, and outputs a reduced Gröbner basis of their kernel intersection. We show, that the proposed KNH interpolation can be used to solve the interpolation step of interpolation-based decoding of interleaved Gabidulin codes in the rank-metric, linearized Reed–Solomon codes in the sum-rank metric and skew Reed–Solomon codes in the skew metric requiring at most \({\widetilde{O}}\left( s^\omega {\mathfrak {M}}(n)\right) \) operations in \({\mathbb {F}}_{q^m}\), where n is the length of the code, \(s\) the interleaving order, \({\mathfrak {M}}(n)\) the complexity for multiplying two skew polynomials of degree at most n, \(\omega \) the matrix multiplication exponent and \({\widetilde{O}}\left( \cdot \right) \) the soft-O notation which neglects log factors. This matches the previous best speeds for these tasks, which were obtained by top–down minimal approximant bases techniques, and complements the theory of efficient interpolation over free skew polynomial modules by the bottom-up KNH approach. In contrast to the top–down approach the bottom-up KNH algorithm has no requirements on the interpolation points and thus does not require any pre-processing.

偏斜多项式是一类非交换多项式，在计算机科学、编码理论和密码学中有多种应用。具体地，偏斜多项式可以用于构造和解码多种度量的评估代码，例如汉明、秩、和秩和偏斜度量。我们提出了 Kötter–Nielsen–Høholdt (KNH) 插值算法的快速分治变体：它输入倾斜多项式向量上的线性函数列表，并输出其核交集的简化 Gröbner 基。我们表明，所提出的 KNH 插值可用于解决秩度量中的交错 Gabidulin 码、和秩度量中的线性化 Reed-Solomon 码以及最多需要\({\widetilde{O}}\left( s^\omega {\mathfrak {M}}(n)\right) \) 运算的偏斜度量 \({\mathbb { F }}_{q ^m}\)，其中n是代码长度，\(s\)交织阶数，\({\mathfrak {M}}(n)\)两个次数最多为n的倾斜多项式相乘的复杂度、\(\omega \)矩阵乘法指数和\({\widetilde{O}}\left( \cdot \right) \)忽略对数因子的软O表示法。这与之前通过自上而下的最小逼近基技术获得的这些任务的最佳速度相匹配，并补充了通过自下而上的 KNH 方法对自由偏斜多项式模块进行有效插值的理论。与自上而下的方法相比，自下而上的 KNH 算法对插值点没有要求，因此不需要任何预处理。