# Discrete weighted transforms and large-integer arithmetic

@article{Crandall1994DiscreteWT,
title={Discrete weighted transforms and large-integer arithmetic},
author={Richard E. Crandall and Barry S. Fagin},
journal={Mathematics of Computation},
year={1994},
volume={62},
pages={305-324}
}
• Published 1994
• Mathematics, Computer Science
• Mathematics of Computation
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…
94 Citations

### Integer multiplication in time O(n log n)

• Computer Science, Mathematics
Annals of Mathematics
• 2021
An algorithm is presented that computes the product of two n-bit integers in O(n log n) bit operations, thus confirming a conjecture of Schonhage and Strassen from 1971, and using a novel “Gaussian resampling” technique that enables the integer multiplication problem to be reduced to a collection of multidimensional discrete Fourier transforms over the complex numbers.

### Parallel Algorithm for Multiplying Integer Polynomials and Integers

Under certain conditions the authors' integer polynomial multiplication method is asymptotically faster than the algorithm based on Fast Fourier Transform when applied to multiply both: polynomials and their coefficients.

### Some Parallel Algorithms for Integer Factorisation

This work describes several integer factorisation algorithms, considers their suitability for implementation on parallel machines, and gives examples of their current capabilities.

### Spectral arithmetic in Montgomery modular multiplication

• Computer Science, Mathematics
Journal of Cryptographic Engineering
• 2017
This survey paper introduces the development of spectral-based MMM, as well as its two important properties: high parallelism and low complexity, and compares these algorithms in terms of digit-level complexity.

### Fast convolutions meet Montgomery

This paper gives a method for understanding and bypassing the short multiplication problem, thus reducing the costs of ring arithmetic to roughly 2M(R) when also using fast convolutions.

### Integer convolution via split-radix fast Galois transform

The fact of real-valued, integer input signals can be exploited along such lines, to enhance the performance of a resulting fast Galois transform (FGT) algorithm, and split-radix FFT structure can be bestowed upon the FGT, boosting eciency for integer convolution.

### Parallel Implementation of Multiple-Precision Arithmetic and 1 , 649 , 267 , 440 , 000 Decimal Digits of π Calculation

A parallel implementation of floating-point real FFT-based multiplication is used and an operation of releasing propagated carries and borrows in multiple-precision addition, subtraction and multiplication is parallelized.

### Fast , Parallel Algorithm for Multiplying Polynomials with Integer Coefficients

Under certain conditions, the algorithm designed specifically to accelerate the process of generating modular polynomials is asymptotically faster than the algorithm based on Fast Fourier Transform when applied to multiply both: numbers and polynmials.

### Faster integer and polynomial multiplication using cyclotomic coefficient rings

• Computer Science, Mathematics
ArXiv
• 2017
An algorithm that computes the product of two n-bit integers in O(n log n (4\sqrt 2)^{log^* n}) bit operations is presented and it is proved that for a fixed prime p, polynomials in F_p[X] of degree n may be multiplied in O (nlog n 4-bit n) bit operations.

## References

SHOWING 1-10 OF 18 REFERENCES

### Computational Complexity of Fourier Transforms over Finite Fields

• Computer Science, Mathematics
• 1977
This method for computing the Discrete Fourier Transform (DFT) of a sequence of n elements over a finite field GF with a number of bit operations 0(nm log (nm).

### Large Integer Multiplication on Hypercubes

• B. Fagin
• Computer Science
J. Parallel Distributed Comput.
• 1992

### The use of finite fields to compute convolutions

• Mathematics
IEEE Trans. Inf. Theory
• 1975
If q is a Mersenne prime, one can utilize the fast Fourier transform (FFT) algorithm to yield a fast convolution without the usual roundoff problem of complex numbers.

### Real-valued fast Fourier transform algorithms

• Computer Science
IEEE Trans. Acoust. Speech Signal Process.
• 1987
A new implementation of the real-valued split-radix FFT is presented, an algorithm that uses fewer operations than any otherreal-valued power-of-2-length FFT.

### The fast Fourier transform in a finite field

A transform analogous to the discrete Fourier transform may be defined in a finite field, and may be calculated efficiently by the 'fast Fourier transform' algorithm. The transform may be applied to

### IRREGULAR PRIMES TO ONE MILLION

• Mathematics
• 1992
Using "fast" algorithms for power series inversion (based on the fast Fourier transform and multisectioning of power series), we have calculated all irregular primes up to one million, including

### Speeding the Pollard and elliptic curve methods of factorization

Since 1974, several algorithms have been developed that attempt to factor a large number N by doing extensive computations module N and occasionally taking GCDs with N. These began with Pollard's p 1

### A Stochastic Roundoff Error Analysis for the Fast Fourier Transform

We study the accuracy of the output of the Fast Fourier Transform by estimating the expected value and the variance of the accompanying linear forms in terms of the expected value and variance of the

### FIR filtering by the modified Fermat number transform

• Mathematics
IEEE Trans. Acoust. Speech Signal Process.
• 1990
Right-angle circular convolution (RCC) and the modified Fermat number transform (MFNT) are introduced. It is shown that a linear convolution of two N-point sequences can be obtained by a

### The twentieth Fermat number is composite

• Computer Science
• 1988
The authors proved that F20 = 2220 +1, which had been the smallest Fermat number of unknown character, is composite, and this computation, written entirely in Cray Fortran and called Cray library functions for the FFT's, would be impossible to do even on supercomputers without fast Fourier transform techniques for integer multiplication.