Lecture 11: Polynomial and Integer Multiplication using the FFT

Background Material

• The material on interpolation and a modular algorithm for polynomial multiplication from Lecture 9 and Lecture 10 on the FFT.
• Elements of Algebra and Algebraic Computing, John D. Lipson, Benjamin Cummings Publishing Co., 1981.

Reading

• Chapter 9 (pp. 293-308) of Lipson (handout, see above for reference).

Topics

1. Review of evaluation, pointwise product, interpolation algorithm for polynomial multiplication.
2. The inverse DFT.
3. Computation of the FFT in Z_p.
   1. The primitive element theorem.
   2. The prime number theorem and Fourier primes.
4. The three primes algorithm.

Lecture Slides

1. fast.ppt - power point slides containing lecture notes on FFT-based polynomial and integer (3 primes) multiplication.

Maple worksheets and programs

• interp.mws - Worksheet containing an implementation of the evaluate, pointwise product, interpolate algorithm for polynomial multiplication.
• fft.mw - Worksheet containing an implementation of a recursive FFT.

Assignment

• Assignment 3