Multiplication Algorithm of Large Integers using Finite Discrete Convolution

Abstract

Several existing algorithms for multiplication of large integers are discussed, and a highly efficient algorithm based on finite discrete convolution is introduced. In the new algorithm, large integers are split into many digits and stored in arrays, every item in array stands for every digit of large integer. In this way, the integer can be any large; the only limit is the memory of computer. The result of finite discrete convolution of the sequence of large integers needs a simple process to become the multiplication result of large integers, which is similar to FFT based algorithm. When processing large integers less than 150 digits, the time cost is less than 1 millisecond. When the large integer is larger than 300 digits, the time cost is only about 3 milliseconds. Even when the original integers become nearly 50000 digits long, the multiplication only needs about 40 seconds. In application, such integers are large enough to process most cases. Algorithm analysis and lots of experimental results show that this algorithm based on finite discrete convolution is much more efficient and is of great significance. More important, it points out a new research direction of finite discrete convolution.