Discrete weighted transforms and large-integer arithmetic

Abstract

It is well known that Discrete Fourier Transform (DFT) techniques may be used to multiply large integers. We introduce the concept of Discrete Weighted Transforms (DWTs) which, in certain situations, substantially improve the speed of multiplication by obviating costly zero-padding of digits. In particular, when arithmetic is to be performed modulo Fermat Numbers 2 2 m + 1 {2^{{2^m}}} + 1 , or Mersenne Numbers 2 q − 1 {2^q} - 1 , weighted transforms effectively reduce FFT run lengths. We indicate how these ideas can be applied to enhance known algorithms for general multiplication, division, and factorization of large integers.