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

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

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
# additions Add(n) = 2· Add(n/2) + n = n log n
# multiplications Mult(n) = 2· Mult(n/2) + n/ = 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 of Then This 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 is c = (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,
Theorem is 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,b define 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 and Applications to Multiplication
Analysis of Algorithms
Prepared by
John Reif, Ph.D.