The Fast Fourier Transform and Applications to Multiplication

1. The Fast Fourier Transform andApplications to Multiplication Analysis of Algorithms Prepared by John Reif, Ph.D.

2. Topics and Readings: - The Fast Fourier Transform Advanced Material :- Using FFT to solve other Multipoint Evaluation Problems- Applications to Multiplication • Reading Selection: • CLR, Chapter 30

3. Nth Roots of Unity • Assume Commutative Ring (R,+,·, 0,1) • ω is principal nth root of unity if • ωk ≠ 1 for k = 1, …, n-1 • ωn = 1, and • Example: for complex numbers

4. Example of nth Root of Unity for Complex Numbers is the 8th root of unity

5. Fourier Matrix

6. Discrete Fourier Transform Input a column n-vector a = (a0, …, an-1)T Output an n-vector which is the product of the Fourier matrix times the input vector

7. Inverse Fourier Transform

8. Fourier Transform is Polynomial Evaluation at the Roots of Unity Input a column n-vector a = (a0, …, an-1)T Output an n-vector (f0, …, fn-1)T which are the values polynomial f(x)at the n roots of unity

9. Fast Fourier Transform • Viewed as Evaluation Problem: naïve algorithm takes n2 ops • Divide and Conquer gives FFT with O(n log n) ops for n a power of 2 • Key Idea: • If ω is nth root of unity then ω2 is n/2th root of unity • So can reduce the problem to two subproblems of size n/2

10. Algorithm FFTn • Input a = (a0, …, an-1)T, n a power of 2

11. FFT Circuit (also known as Butterfly Network) • Total Recursion depth = log n • Communication Distance 2d at depth d

14. Operation Counts for FFT Algorithm • Assume n = 2k • # additionsAdd(n) = 2· Add(n/2) + n = n log n • # multiplicationsMult(n) = 2· Mult(n/2) + n/2 = ½ n log n • Total Time O(n log n) • Note in complex FFT, # real ops is 5 n log n

15. Multipoint Polynomial Evaluation • Input polynomial • Problem evaluate f(x) at x0, x1, …, xn-1 • Easy Cases: eval at roots of unity FFT Case xk =ωk for k=0,…,n

16. Multipoint Polynomial Evaluation (cont’d)

17. Other Polynomial Evaluation Problems Solved by FFT Each costs O(n log n) time • Evaluate at points Xi = bai + d for i=0,…, n-1(Chirp Transfom) • Reduced to FFT • Single point evaluation of all derivatives of a polynomial • Solve by reduction to above Chirp Transform of case 2) • Evaluate at points Xi = b(ai)2+ cai + d for i=0,…, n-1 • Solve by divide and conquer similar to FFT

18. Single Point Evaluation of all Derivatives of Polynomial • Input and point x0 • output

19. Single Point Evaluation of all Derivatives of Polynomial (cont’d) • Taylor Series Representation ofThenThis reduces to case of evaluation at points • Solve this Chirp Transform problem by reduction FFT

20. Advanced Material: Further Applications of FFT • Convolution: Products and Powers of Polynomials • Used for for Integer Multiplication Algorithms • Also used for Filtering on infinite input streams • Division and Inverse of Polynomials • Multipoint Evaluation and Interpolation

21. Advanced Material: Products and Powers of Polynomials • Input vectors a = (a0, a1, …, an-1)T b = (b0, b1, …, bn-1)T • Definition of Convolution c = a ⊗b Where for i=0, …, 2n-1 define ak = bk = 0 if k< 0 or k>n

22. Products and Powers of Polynomials (cont’d) • Convolution Theorem • Application to Polynomial Products:

23. Products of m Polynomials • Generalized Convolution Theorem

24. Wrapped Convolutions • a = (a0, a1, …, an-1)T , b = (b0, b1, …, bn-1)T • Positive wrapped convolution isc = (c0, c1, …, cn-1)T • Negative wrapped convolution is d = (d0, d1, …, dn-1)T

25. Application of Wrapped Convolution to Modular Polynomial Products

26. Computing Positive Wrapped Convolution • Let ω = principal nth root of unity • Assume n has multiplicative inverse, Theoremis the positive wrapped convolution of n-vectors a and b.

27. Computing Negative Wrapped Convolution • Also is the negatively wrapped convolution of n-vectors a and b where and Ψ2 = ω = principal nth root of unity

28. Integer Multiplication by Polynomial Product (solved via FFT) • Input n bit integers a,bdefine polynomials degree k = n/L

29. Integer Multiplication by Polynomial Product (cont’d) • Idea • Compute c(x) = a(x)· b(x) by convolution • Evaluate c(2L) = a· b

30. Integer Multiplication Algorithms using Reduction to Polynomial Product • Pollard Mult Algorithm • Karp Mult Algorithm • Schönage-Strassen Mult Algorithm

31. Pollard Multiplication Algorithm • n = kL, L = 1 + log k • Choose primes P1, P2, P3 where • Compute C(x) by convolution over finite field Zpi for i =1,2,3 (requires k mults on 2L bit integers)

32. Pollard Multiplication Algorithm (cont’d) • Evaluate C(2L) • Time Bounds recursive mults FFT

33. Korp Multiplication Algorithm • Compute C(x) modulo k by convolution • Compute C(x) modulo (22L+1) by convolution • Compute C(x) coefficients from C(x) mod k, C(x) mod (22L+1) by Chinese remaindering

34. Korp Multiplication Algorithm (cont’d) • Compute C(2L) • Time recursive mults FFT

35. Schönage-Strassen Multiplication Algorithm (2’) Compute C(x) mod (xk+1) modulo (22L+1) by wrapped convolution Requires only k recursive mults on 2L bit numbers • Time recursive mults FFT

36. Still Open Problem: How Fast Can You Multiply Integers? • Can you mult n bit integers in O(n log n) time?

37. The Fast Fourier Transform andApplications to Multiplication Analysis of Algorithms Prepared by John Reif, Ph.D.