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… Expand
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References

SHOWING 1-10 OF 18 REFERENCES
Computational Complexity of Fourier Transforms over Finite Fields
Abstract : This paper describes a method for computing the Discrete Fourier Transform (DFT) of a sequence of n elements over a finite field GF (p to the mth power) with a number of bit operationsExpand
Large Integer Multiplication on Hypercubes
  • B. Fagin
  • Computer Science
  • J. Parallel Distributed Comput.
  • 1992
TLDR
A convolution algorithm on a massively parallel processor, based on the Fermat Number Transform, is presented and some examples of the trade-offs between modulus, interprocessor communication steps, and input size are presented. Expand
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. Expand
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. Expand
Parameter determination for complex number-theoretic transforms using cyclotomic polynomials
Some new results for finding all convenient moduli m for a complex numbertheoretic transform with given transform length n and given primitive nth root of unity modulo m are presented. The mainExpand
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 toExpand
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, includingExpand
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 1Expand
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 theExpand
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 aExpand
...
1
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