On Short Multiplications and Divisions

Abstract.

Computing only the low degree terms of the product of two univariate polynomials is called a short multiplication. By decomposition into subproblems, a short multiplication can be reduced to appropriate addition of the results of a number of full multiplications. In this paper a new way of choosing the size of the subproblems is proposed. Computing the quotient of two polynomials is called a short division. The ideas used in the short multiplication algorithm are transferred to an algorithm for short divisions. Finally, several applications of short multiplications and divisions are pointed out.