Design and Implementation of Long-Digit Karatsuba’s Multiplication Algorithm Using Tensor Product Formulation

Karatsuba’s multiplication algorithm uses three singledigit multiplications to perform one two-digit multiplication. If we apply Karatsuba’s multiplier recursively, it takes only 3 single-digit multiplications to multiply a pair of 2-digit numbers. This is a significant improvement compared to 4 single-digit multiplications using grade-school multiplier. In this paper, we will use tensor production formulation to express Karatsuba’s multiplication algorithm in both recursive and iterative form. Usually, Karatsuba’s algorithm is implemented as recursive program. With the iterative tensor product formula of Karatsuba’s algorithm, we can derive an iterative (for loop) program to perform multiplication of long-digit numbers. Furthermore, the 3 single-digit multiplications can be fully parallelized. Keyword: tensor product, block recursive algorithm, programming methodology, Karatsuba’s multiplication algorithm.