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Fast Multiplication of Large Numbers Using Fourier Techniques

Learn about fast multiplication of large numbers using Fourier techniques, starting with basic multiplication and progressing to polynomial multiplication and convolution. Explore how the Fast Fourier Transform (FFT) can be utilized for efficient multiplication, compare FFT multiplication with regular convolution in terms of time complexity, and discover the benefits of this method for multiplying large numbers quickly.

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Fast Multiplication of Large Numbers Using Fourier Techniques

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  1. Fast Multiplication of Large Numbers Using Fourier Techniques Henry Skiba Advisor: Dr. Marcus Pendergrass

  2. What is Multiplication? • Multiplication of small numbers • This can also be represented in the following way: • where x is equal to 10 • Multiplication of small numbers

  3. Multiplication in a New Way • Represent basic numbers as polynomials when x equals 10 • Multiply two polynomials together

  4. General Polynomial Multiplication • General Polynomials • General Polynomial Multiplication • Even More General Form • where • Convolution

  5. What is Convolution? • This opens the door for use of the Fast Fourier Transform (FFT) for multiplication

  6. Using the FFT to Convolve • Convolution in the time domain is point wise multiplication in the frequency domain • But

  7. Using the FFT to Convolve (cont.) • Arrays are zero padded to make and equivalent • p = [p0,p1,p2,…,pn-1,pn,0,0,0,0,0…] vs. with zero padding without zero padding

  8. Algorithm • Starts with two arrays, p and q • Zero pads p and q • carry

  9. Testing vs. Regular Convolution in Time Domain • Timed how long it took to multiply to random numbers of a certain length • 10,000 to 100,000 with digit stepping of 10,000 • Repeated iteration 5 times • Used theoretical times to plot “fit” based on average of 10,000 digit time values • FFT Multiplication: nlog10n • Convolution: n2

  10. Results

  11. Questions?

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