A new Las Vegas algorithm is presented for the composition of two polynomials modulo a third one, over an arbitrary field. When the degrees of these polynomials are bounded by $n$, the algorithm uses $O(n^{1.43})$ field operations, breaking through the $3/2$ barrier in the exponent for the first time. The previous fastest algebraic algorithms, due to Brent and Kung in 1978, require $O(n^{1.63})$ field operations in general, and ${n^{3/2+o(1)}}$ field operations in the special case of power series over a field of large enough characteristic. If cubic-time matrix multiplication is used, the new algorithm runs in ${n^{5/3+o(1)}}$ operations, while previous ones run in $O(n^2)$ operations. Our approach relies on the computation of a matrix of algebraic relations that is typically of small size. Randomization is used to reduce arbitrary input to this favorable situation.

翻译：提出了一种新的拉斯维加斯算法，用于在任意域上对两个多项式模第三个多项式进行复合。当这些多项式的次数受限于$n$时，该算法使用$O(n^{1.43})$次域运算，首次突破了指数上的$3/2$屏障。此前最快的代数算法（由Brent和Kung于1978年提出）一般情况下需要$O(n^{1.63})$次域运算，在特征足够大的域上的幂级数特例中需要${n^{3/2+o(1)}}$次域运算。若使用三次时间矩阵乘法，新算法运行时间为${n^{5/3+o(1)}}$次运算，而此前算法需$O(n^2)$次运算。我们的方法依赖于计算一个通常较小的代数关系矩阵。通过随机化将任意输入约简到这种有利情形。