We present a simple and fast algorithm for computing the $N$-th term of a given linearly recurrent sequence. Our new algorithm uses $O(\mathsf{M}(d) \log N)$ arithmetic operations, where $d$ is the order of the recurrence, and $\mathsf{M}(d)$ denotes the number of arithmetic operations for computing the product of two polynomials of degree $d$. The state-of-the-art algorithm, due to Charles Fiduccia (1985), has the same arithmetic complexity up to a constant factor. Our algorithm is simpler, faster and obtained by a totally different method. We also discuss several algorithmic applications, notably to polynomial modular exponentiation, powering of matrices and high-order lifting.

中文翻译：

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一种计算线性循环序列第 $N$ 项的简单快速算法

我们提出了一种简单而快速的算法来计算给定线性循环序列的第 $N$ 项。我们的新算法使用 $O(\mathsf{M}(d) \log N)$ 算术运算，其中 $d$ 是递归的顺序，$\mathsf{M}(d)$ 表示算术次数计算两个多项式 $d$ 的乘积的操作。最先进的算法，归功于 Charles Fiduccia (1985)，具有相同的算术复杂度，直到一个常数因子。我们的算法更简单、更快，并且是通过完全不同的方法获得的。我们还讨论了几种算法应用，特别是多项式模幂、矩阵幂和高阶提升。