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Lecture 4
Noise as a random number
generator
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Plan of the Lecture
Extracting randomness from physical noise
1 Technical definition of noise
2-3 Thermal noise
4 Randomness from noise
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MATHEMATICS OF N
Many frequencies
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Frequency spectra
1 1 2 1
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0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
M=10
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
M=40
Adding frequencies
0
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NoiseMore frequencies in the spectrum more noisy
0 1000 2000 3000 4000 5000 6000 7000 8000 9-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Little correlation (short memory) hard to predict
In fact, many natural source of noise have a continuous spectru
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Spectral bandwidth vs. memor
Correlation time
Coherence time
Spectrin freq
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THERMAL
Theory
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A resistor
RI
Really zero?
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Contact with a thermal bat
R
Temperature T
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Thermal noise
V
0
V
t
time
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Johnson-Nyquist noise
The thermal noise of the resistor is named af
John Johnson, who reported it, and Harry Nywho did the theoretical description (Bell Labs
From Johnsons paper:
The mean-square potential fIuctuation over the
conductor is proportionalto the electrical resistaand the absolute temperature of the conductor.
independentof the size, shape or material of th
conductor.
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The formula
RPower dissipated
by the resistor
At equilibrium, they must be equal
Ban
the Cable = 1D
waveguide for E,B
Boltzmanns c
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More rigorous derivation
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Predictions in graphs
Frequency s
1kW
1.6 10-13V2
=(0.4 mV)2
T=300K, Df=10kHz
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NOISE IN TH
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Devices
R
Amplifier
Oscilloscope
time
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Observation
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RANDOMNESS FROM
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Statistics of noise
The observed distribution is enough, because we have a c
source of noise: we checked that the noise is the one expected for th
we trust physics that it is a phenomenon too comple
One could put up an antenna and capture unknown signals
look random to us. But they may not be random for others,
unwanted structure in them.
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Observed distribution
R=1kW
T=300K
time Counts (103
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Possible processing (1)
Try to get directly fair coin sequences out o
0 1 000
1
1
0
1
More than one bit need to adapt carefully the in
for each sequence to have the s
(2)
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Possible processing (2)
Remark: for a trusted source, one can be less conservativ
compute randomness from the average (Shannon) entro
C l ti f fi it b d
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Correlations from finite bandwPreviously we assumed that each value of V is independent of t
As we know, no problem in principle: just estimate P of each se
compute Hmin. But in practice, this is not feasible.
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Wikipedia pages: http://en.wikipedia.org/wiki/Hardware_random_number_generat
Suggested Readings
S f L t 4
http://en.wikipedia.org/wiki/Hardware_random_number_generatorhttp://en.wikipedia.org/wiki/Hardware_random_number_generator7/27/2019 Lecture Slides Lecture 4 Lecture4
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Summary of Lecture 4
Noise = large spectrum of frequencies
Thermal noise
Extracting randomness, effects of correlation
Extracting randomness from physical noise