Discrete Weighted Transforms and Large-Integer Arithmetic

Abstract

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 Fermât Numbers 22"1 + 1 , or Mersenne Numbers 29 -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 oflarge integers.

Three new factors of Fermat numbers

“…The effectiveness of the Cruncher increases as the integers to be factored increase in size. However, due to memory limitations, we have not attempted to factor Fermat numbers larger than [19, §4.3.3] or other "fast" algorithms [11,13] are preferable for large w on a workstation. The crossover point depends on details of the implementation.…”

“…He also "rediscovered" the 49-digit factor of F 9 and the five known prime factors of F 12 . Crandall [10] discovered two 19-digit factors of F 13 . This paper reports the discovery of 27-digit factors of F 13 and F 16 (the factor of F 13 was announced in [3, §8]) and of a 33-digit factor of F 15 .…”

“…Crandall [10] discovered two 19-digit factors of F 13 . This paper reports the discovery of 27-digit factors of F 13 and F 16 (the factor of F 13 was announced in [3, §8]) and of a 33-digit factor of F 15 . All three factors were found by ECM, although the implementations and hardware differed between F 13 and F 15 , F 16 .…”

“…This paper reports the discovery of 27-digit factors of F 13 and F 16 (the factor of F 13 was announced in [3, §8]) and of a 33-digit factor of F 15 . All three factors were found by ECM, although the implementations and hardware differed between F 13 and F 15 , F 16 . In fact, we used Dubner Crunchers (see §3) on F n for 12 ≤ n ≤ 14, and Sun workstations with DWT multiplication (see §4.2) for 16 ≤ n ≤ 21, as well as a Pentium Pro for n = 15, again with DWT multiplication.…”

“…In fact, we used Dubner Crunchers (see §3) on F n for 12 ≤ n ≤ 14, and Sun workstations with DWT multiplication (see §4.2) for 16 ≤ n ≤ 21, as well as a Pentium Pro for n = 15, again with DWT multiplication. Details of the computations on F 13 , F 16 and F 15 are given in §5, §6 and §7 respectively.…”