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}
}
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… 

Rapid multiplication modulo the sum and difference of highly composite numbers

TLDR
Tight bounds on the rounding errors which naturally occur in floating-point implementations of FFT and DWT multiplications are proved, making it possible for FFT multiplications to be used in situations where correctness is essential, for example in computer algebra packages.

Integer multiplication in time O(n log n)

TLDR
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

TLDR
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

TLDR
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

TLDR
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

TLDR
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.

Recent Progress and Prospects for Integer Factorisation Algorithms

  • R. Brent
  • Computer Science, Mathematics
    COCOON
  • 2000
TLDR
This paper considers the problem of parallel solution of the large, sparse linear systems which arise with the MPQS and NFS methods, and outlines several integer factorisation algorithms, consider their suitability for implementation on parallel machines, and give examples of their current capabilities.

Integer convolution via split-radix fast Galois transform

TLDR
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.

Area-Time Efficient Architecture of FFT-Based Montgomery Multiplication

TLDR
This paper integrates the fast Fourier transform (FFT) method into the McLaughlin’s framework, and presents an improved FFT-based Montgomery modular multiplication (MMM) algorithm achieving high area-time efficiency.

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

TLDR
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.
...

References

SHOWING 1-10 OF 18 REFERENCES

Large Integer Multiplication on Hypercubes

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

The use of finite fields to compute convolutions

TLDR
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

TLDR
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.

Parameter determination for complex number-theoretic transforms using cyclotomic polynomials

TLDR
The main result is based on the prime factorization for values of cyclotomic polynomials in the ring of Gaussian integers.

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

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

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

TLDR
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.