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The usual formulation of efficient division uses Newton iteration to computean inverse in a related domain where multiplicative inverses exist. On onehand, Newton iteration allows quotients to be calculated using an efficientmultiplication method. On the other hand, working in another domain is notalways desirable and can lead to a library structure where arithmetic domainsare interdependent. This paper uses the concept of a whole shifted inverse andmodified Newton iteration to compute quotients efficiently without leaving theoriginal domain. The iteration is generic to domains having a suitable shiftoperation, such as integers or polynomials with coefficients that do notnecessarily commute.

Efficient Generic Quotients Using Exact Arithmetic