Modular composition is the problem of computing the composition of two univariate polynomials modulo a third one. For a long time, the fastest algebraic algorithm for this problem was that of Brent and Kung (1978). Recently, we improved Brent and Kung's algorithm by computing and using a polynomial matrix that encodes a certain basis of algebraic relations between the polynomials. This is further improved here by making use of two polynomial matrices of smaller dimension. Under genericity assumptions on the input, this results in an algorithm using $\tilde{O}(n^{(ω+3)/4})$ arithmetic operations in the base field, where $ω$ is the exponent of matrix multiplication. With naive matrix multiplication, this is $\tilde{O}(n^{3/2})$, while with the best currently known exponent $ω$ this is $O(n^{1.343})$, improving upon the previously most efficient algorithms.

翻译：模合成问题旨在计算两个单变量多项式在模第三个多项式下的复合。长期以来，该问题最快的代数算法是Brent和Kung（1978）提出的方法。最近，我们通过计算并利用一个编码多项式间特定代数关系基的多项式矩阵，改进了Brent和Kung的算法。本文进一步引入两个维度更小的多项式矩阵，实现了更优的改进。在输入满足一般性假设的条件下，该算法在基域上仅需$\tilde{O}(n^{(ω+3)/4})$次算术运算，其中$ω$为矩阵乘法的指数。若采用朴素矩阵乘法，复杂度为$\tilde{O}(n^{3/2})$；而使用当前已知的最优指数$ω$时，复杂度可降至$O(n^{1.343})$，较此前最高效的算法实现了显著提升。