Discrete weighted transforms and large-integer arithmetic

  title={Discrete weighted transforms and large-integer arithmetic},
  author={Richard E. Crandall and Barry S. Fagin},
  journal={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… Expand
Rapid multiplication modulo the sum and difference of highly composite numbers
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. Expand
Integer multiplication in time O(n log n)
We present an algorithm 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. Our complexity analysis takesExpand
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. Expand
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. Expand
Spectral arithmetic in Montgomery modular multiplication
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. Expand
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. Expand
Recent Progress and Prospects for Integer Factorisation Algorithms
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. Expand
Integer convolution via split-radix fast Galois transform
Integer convolution can be eected, as is well known, via certain number-theoretical transforms. One particular transform, which we call a discrete Galois transform (DGT), can be used eciently forExpand
Area-Time Efficient Architecture of FFT-Based Montgomery Multiplication
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. Expand
Parallel Implementation of Multiple-Precision Arithmetic and 1 , 649 , 267 , 440 , 000 Decimal Digits of π Calculation
We present efficient parallel algorithms for multiple-precision arithmetic operations of more than several million decimal digits on distributed-memory parallel computers. A parallel implementationExpand


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
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
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
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
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