FFT And Fast Multiplication

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FFT And Fast Multiplication

• Robert L. Watson
• 12 December 2006

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Objectives

• Establish Fast Fourier Transform
• Use FFT to establish algorithms for fast integer
  and polynomial multiplication

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Preliminary Definitions
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Algorithm for FFT
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Justification
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FFT
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Application of FFT

• We can compute the convolution of two polynomials
  in a fast manner.
• The convolution of f and g in Rx

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Fast Convolution (CNV)
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Fast Multiplication of Integers

• Use CNV algorithm to quickly compute the product
  of two integers.
• Define polynomials A and B, and compute their
  convolution rem pj,where pj is one of three
  primes
• Chinese Remainder Theorem guarantees AB can be
  reconstructed from the convolution rem pj.

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Fast Multiplication of Integers

• A and B are in 64-ary representation

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Three Primes FFT (CK1)
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Three Primes FFT (CK1)
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Fast Multiplication of Polynomials

• Convolution is equivalent to multiplication in
  Rx / ltxn - 1gt.
• We want to perform multiplication in an arbitrary
  ring R, but FFT has specific requirements for the
  roots of unity.
• Consider virtual roots of unity.

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Fast Multiplication of Polynomials

• Consider R where 2 is a unit.
• Obtain
• This is the negative wrapped convolution of f and
  g.

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Fast Multiplication of Polynomials

• Definitions

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Schonhage and Strassens Multiplication Algorithm
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Fast Multiplication of Polynomials