Transcript Fast Multiplication of Large Numbers Using Fourier Techniques

Fast Multiplication of Large
Numbers Using Fourier
Techniques
0011 0010 1010 1101 0001 0100 1011
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Henry Skiba
Advisor: Dr. Marcus Pendergrass
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What is Multiplication?
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• Multiplication of small numbers
123
123
 654
400 80  12
 654
5000 1000 150
 60000 12000 1800
492
6150
 73800
80442
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12x0  (8  15) x  (4  10  18) x 2  (5  12) x3  6x 4
– where x is equal to 10
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• This can also be represented in the
following way:
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Multiplication in a New Way
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• Represent basic numbers as
polynomials when x equals 10
123 p( x)  3x0  2x  1x 2
654  q( x)  4 x0  5x  6 x 2
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• Multiply two polynomials together
p( x)  q( x)  (3  4) x0  (3 5  4  2)x  (3  6  2  5  1 4) x 2  (2  6  1 5) x3  (1 6) x 4
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General Polynomial Multiplication
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• General Polynomials
p( x)  p0 x 0  p1 x  p2 x 2  ...  pn1 x n 1  pn x n
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q( x)  q0 x 0  q1 x  q2 x 2  ...  qm1 x m1  qm x m
• General Polynomial Multiplication
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r( x)  p( x)  q( x)  ( p0q0 ) x  ( p1q0  q1 p0 ) x  ( p2q0  p1q1  p0q2 ) x  ... ( pn1qm  pn qm1 ) xmn1  ( pn qm ) xmn
0
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• Even More General Form
–
m n
r ( x)   ri x i
i 0
• Convolution
i
where ri   pk qi k
k 0
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What is Convolution?
0011 0010 1010 1101 0001 0100 1011

• s  h(t )   s(u)h(t  u)du

•
i
ri  p  qi   pk qi k
k 0
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• This opens the door for use of the Fast
Fourier Transform (FFT) for
multiplication
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Using the FFT to Convolve
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• Convolution in the time domain is point
wise multiplication in the frequency
domain
p FFT

 P
q 
 Q
FFT
p  q  FFT1 ( P  Q)
• But
1
FFT ( P  Q)  p  q
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Using the FFT to Convolve (cont.)
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• Arrays are zero padded to make  and 
equivalent
– p = [p0,p1,p2,…,pn-1,pn,0,0,0,0,0…]
000123
045600
with zero
padding
vs.
124123
645645
without zero
padding
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Algorithm
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Starts with two arrays, p and q
Zero pads p and q
p, q FFT

 P, Q
P  Q Multiply
 PQ
FFT 1
PQ  p  q
carry p  q
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Testing vs. Regular Convolution
in Time Domain
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• Timed how long it took to multiply to random
numbers of a certain length
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– 10,000 to 100,000 with digit stepping of 10,000
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• Repeated iteration 5 times
• Used theoretical times to plot “fit” based on
average of 10,000 digit time values
– FFT Multiplication: nlog10n
– Convolution: n2
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Results
0.11
0.1
0.09
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0.08
0.07
0.06
60
0.05
0.04
50
0.03
0.02
0.01
40
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2
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4
30
2
10
20
1
10
10
0
10
0
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10
-1
10
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x 10
-2
10
-3
10
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1
5
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7
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8
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7
10
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x 10
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9
8
10
9
10
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x 10
Questions?
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