This article describes and analyzes all existing algorithms for computing $x=a^{-1}\pmod {p^k}$ x = a - 1 ( mod p k ) for a prime $p$ p , and also introduces a new algorithm based on the exact solution of linear equations using $p$ p -adic expansions. The algorithm starts with the initial value $c=a^{-1}\pmod {p}$ c = a - 1 ( mod p ) and iteratively computes the digits of the inverse $x=a^{-1}\pmod {p^k}$ x = a - 1 ( mod p k ) in base $p$ p . The mod 2 version of the algorithm is more efficient than all existing algorithms for small values of $k$ k . Moreover, it stands out as being the only one that works for any $p$ p , any $k$ k , and digit-by-digit. While the new algorithm is asymptotically worse off, it requires the minimal number of arithmetic operations (just a single addition) per step, as compared to all existing algorithms.

中文翻译：

---

Inversion mod $p_k$ 的算法

本文描述和分析了所有现有的计算 $x=a^{-1}\pmod {p^k}$ x = a - 1 ( mod pk ) 的素数 $p$ p 的算法，并介绍了一种新算法基于使用 $p$ p -adic 展开式的线性方程的精确解。算法从初始值 $c=a^{-1}\pmod {p}$ c = a - 1 ( mod p ) 开始，迭代计算逆数 $x=a^{-1}\pmod {p^k}$ x = a - 1 ( mod pk ) 以 $p$ p 为基数。对于 $k$ k 的小值，该算法的 mod 2 版本比所有现有算法更有效。此外，它是唯一适用于任何 $p$ p 、任何 $k$ k 和逐位数字的方法。尽管新算法的性能逐渐变差，但与所有现有算法相比，它每步需要最少的算术运算（仅一次加法）。