The usual formulation of efficient division uses Newton iteration to compute an inverse in a domain where multiplicative inverses exist. On one hand, this allows quotients to be calculated using an efficient multiplication method. On the other hand, working in another domain is not always desirable and can lead to a library structure where arithmetic domains are interdependent. This paper uses the concept of a shifted inverse and modified Newton iteration to compute quotients efficiently without leaving the original domain. The iteration is generic to Euclidean domains having a suitable shift operation, such as base-$B$ integers or polynomials in a monomial basis.

翻译：传统高效除法的实现通常采用牛顿迭代法，在存在乘法逆元的域内计算倒数。一方面，这种方法允许通过高效乘法算法求取商值；另一方面，跨域操作并非总是最优选择，可能导致算术域相互依赖的库结构问题。本文提出基于移位逆元和修正牛顿迭代法的技术方案，可在不脱离原始域的前提下高效计算商值。该迭代方法适用于具有合适移位运算的欧几里得域，例如基-$B$整数或单项式基多项式域。