Related Concepts
Definition
A method for multiplying two polynomials that saves coefficient multiplications at the cost of extra additions compared to the schoolbook multiplication method.
Theory
The Karatsuba Algorithm (KA) for multiplying two polynomials was introduced in 1962 [3]. It saves coefficient multiplications at the cost of extra additions compared to the schoolbook or ordinary multiplication method. The basic KA is performed as follows. Consider two degree-1 polynomials A(x) and B(x) with n = 2 coefficients.
$$A(x) = {a}_{1}x + {a}_{0}$$
$$B(x) = {b}_{1}x + {b}_{0}$$
Let \({D}_{0},{D}_{1},{D}_{0,1}\) be auxiliary variables with
$${D}_{0} = {a}_{0}{b}_{0}$$
$${D}_{1} = {a}_{1}{b}_{1}$$
$${D}_{0,1} = ({a}_{0} + {a}_{1})({b}_{0} + {b}_{1})$$
Then the polynomial \(C(x) = A(x)B(x)\) can be calculated in the following way:
$$C(x) = {D}_{1}{x}^{2} + ({D}_{ 0,1} - {D}_{0} - {D}_{1})\ x + {D}_{0}$$
This method requires three multiplications and four...