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

2. 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. 3 Represent basic numbers as polynomials when x equals 10 Multiply two polynomials together Multiplication in a New Way

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

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

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

7. 7 Arrays are zero padded to make and equivalent –p = [p 0,p 1,p 2,…,p n-1,p n,0,0,0,0,0…] Using the FFT to Convolve (cont.) vs. with zero padding without zero padding

8. 8 Algorithm 1.Starts with two arrays, p and q 2.Zero pads p and q 3. 4. 5. 6.carry

9. 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: nlog 10 n –Convolution: n 2

10. 10 Results

11. 11 Questions?