Lecture: The Fast Fourier Transform and Fast Polynomial and Integer Arithmetic

Background Material

• Lecture on Karatsuba's algorithm
• Lecture on modular arithmetic and modular algorithms

Reading

• John D. Lipson, Elements of Algebra and Algebraic Computing, Addison-Wesley, 1981, Chapter 8, pp. 293-312.

Topics

1. Modular algorithms
   1. Polynomial multiplication: 1) evaluate, 2) pointwise product, 3) interpolate
   2. Integer multiplication: 1) reduce mod primes, 2) modular products, 3) CRA.
2. Discrete Fourier transform (DFT)
3. Fast Fourier Transform (FFT)
4. FFT based polynomial multiplication
5. Modular FFTs
6. Three primes algorithm for integer multiplication

Slides and Worksheets

1. interp.mw - worksheet with modular algorithms for integer and polynomial multiplication
2. Slides reviewing complex arithmetic (complex.ppt, complex.pdf)
3. Slides on the FFT (fft.ppt,fft.pdf)
4. fft.mw - worksheet with FFT implementation
5. Slides on Modular FFTs and the Three Primes Algorithm (threeprimes.ppt, threeprimes.pdf)
6. threeprimes.mw - worksheet with Three Primes Algorithm implementation