1. Communication costs of Schönhage-Strassen fast integer multiplication
Derrick Coetzee
This work released under Creative Commons Zero Waiver by its author, Derrick Coetzee

2. Schönhage-Strassen Algorithm
Classical FFT-based integer multiplication algorithm
O(n log n log log n) on n-bit inputs
Asymptotically fastest algorithm used in practice

3. Acyclic convolution Vector operation “Multiplication without carrying”
Can compute efficiently with convolution theorem
_________

4. Using convolution to multiply
1234
5678
Split and zero pad
Split and zero pad 4 3 2 1 8 7 6 5
DFT
DFT
Recursive pointwise multiplication
IDFT 32 52 61 60 34 16
5______ 32 52 61 60 34 16 5
Recombination (carrying)

5. Vector length and entry size
Take n-bit input, split into 𝑛 parts of 𝑛 bits each
Vector length = 𝑛
Vector entries must be large enough to hold sum of 𝑛 values, each a product of two 𝑛 -bit numbers
⌈lg 𝑛 ( 2 𝑛 −1)2⌉ = 2 𝑛 + ½lg n bits 01 11 10 01
m=2
d = 4
_________

6. Arithmetic cost
All ops except FFT and recursive multiplications are O(n)
Avoid recursive multiplications in FFTs by performing all operations in ℤ2n′+1
FFTs take O(n log n) time
T(n) = 𝑛 T(2 𝑛 + ½lg n) + O(n log n)
Solution: T(n) = O(n log n log log n)

7. Communication cost Suppose each FFT done independently
Hong-Kung FFT lower bound (met by algorithm of Frigo): O(n log n/log M)
Stop recursion when subproblem fits in cache (Q(n) = n for n < αM)
Q(n) = 𝑛 Q(2 𝑛 ) + O(n log n/log M)
Q(n) = O(n (log n/log M) log(log n/log M))