random numbers certified by bells theorem
DESCRIPTION
A brief talk about a paper published by a bunch of people in 2010 on how it is possible to use create devices for generating random numbers in such a manner that anyone can verify that a pseudo-random number generator is not being used, but instead that quantum noise is being used.TRANSCRIPT
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Random Numbers Certified by Bell's Theorem by Pironio and 10 other authors
from 6 different universities and institutions.
A brief introduction by David Kemp
Licensed under a Creative Commons Attribution 3.0 Licensehttp://creativecommons.org/licenses/by/3.0/
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Why?
Pseudo random numbers may have patterns known to attackers.
There is need for a source of randomness that can be verified to be genuinely random.
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Bell’s Inequality
• Published by John Stewart Bell in 1964• Proved “reality not separable from observation”.• Einstein died in 1955 believing reality must be
separable from observation (see EPR paradox).
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Simple Inequality Example
Room full of people:•Only one right-handed male.•Only one blue-eyed female.•How many (max) blue-eyed right-handers?– At most one of them can be male.– At most one of them can be female.– Hence a maximum of 2.
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Simple Inequality as Venn Diagram
Female
Right-handed
Blue-eyed
Size(blue-eyed right-handers) ≤ Size(blue-eyed females) + Size(right-handed males)
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In Probabilistic Terms
Pr(blue-eyed right-hander) ≤Pr(blue-eyed female) + Pr(right-handed male)
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What if your data is incomplete?
• Suppose you can only ask each person one of:– Are you a blue-eyed right-hander?– Are you a blue-eyed female?– Are you a right-handed male?
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What if your data is incomplete?
•Suppose you can only ask each person one of:–Are you a blue-eyed right-hander?–Are you a blue-eyed female?–Are you a right-handed male?
Pr(blue-eyed right-hander) ≤Pr(blue-eyed female) + Pr(right-handed male)
ONLY IF THEY DO NOT LIE!
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Identical Twins
• Second room with a twin of everyone in first.• No communication between the rooms.• Ask every person only one of these questions:– Are you blue-eyed?– Are you left-handed?– Are you female?
• Sometimes ask both twins the same question.• Only way to avoid lie detection is prior agreement.
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Bell’s Inequality
IF
•A, B and C are binary properties,•A', B' and C' are properties of twin,• Pr(x & not(x')) = 0.
THEN
Pr(A & not(B')) + Pr(B & not(C')) ≥ Pr(A & not(C'))
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Pairs of Magic Boxes
A B C A B C
Three buttons and two lights on each box
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Magic Box PairsA B C
Can only press one button on each box.
A B C
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Magic Box Properties
Whenever same button is pressedon both boxes:
Same light flashes
A B C A B C
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•Pr(A:red & B:yellow) ≈ 7.5%•Pr(B:red & C:yellow) ≈ 7.5 %•Pr(A:red & C:yellow) ≈ 25%
Pr(A:red & B:yellow) + Pr(B:red & C:yellow)< Pr(A:red & C:yellow)
A B C A B CMagic Box Properties
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But surely that is impossible!!!
Image by Wild Guru Larry. Some rights reserved.
http://www.flickr.com/photos/wentzelepsy/4435803492/
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Relies on Quantum Entanglement
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Entangled Electrons
• Measure Electron Spin Direction• Spin-up/down instead of clockwise/anti-clockwise• Result seems to be completely random.• Can measure from different angles.–Eg. Vertically, horizontally, and 45º to horizon.
• Measurement disturbs the electron.–Cannot reliably measure same electron from different angles.
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Entangled Electrons
Let buttons A, B and C, measure at 0º, 45º and 90º•Spin-up = red•Spin-down = yellow
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Entangled Electrons
Measure entangled electrons from same direction:
100% Correlation
(Press same button on both boxes and lights agree)
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Entangled Electrons
• A:red & C:red 25%• A:yellow& C:yellow 25%• A:red & C:yellow 25%• A:yellow & C:red 25%
Measure entangled electrons at 90º to each other:50% Correlation
Pr(A:red & B:yellow) + Pr(B:red & C:yellow)< Pr(A:red & C:yellow)
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Entangled Electrons
• A:red & B:red ≈ 42.5%• A:yellow& B:yellow ≈ 42.5%• A:red & B:yellow ≈ 7.5%• A:yellow & B:red ≈ 7.5%
Measure entangled electrons at 45º to each other:≈ 85% Correlation
Pr(A:red & B:yellow) + Pr(B:red & C:yellow)< Pr(A:red & C:yellow)
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Entangled Electrons
• B:red & C:red ≈ 42.5%• B:yellow& C:yellow ≈ 42.5%• B:red & C:yellow ≈ 7.5%• B:yellow & C:red ≈ 7.5%
Measure entangled electrons at 45º to each other:≈ 85% Correlation
Pr(A:red & B:yellow) + Pr(B:red & C:yellow)< Pr(A:red & C:yellow)
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Quantum Entanglement is the only known process that
can violate Bell’s Inequality.
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Verifiably Random
We can use these magic boxes to
generate verifiably random numbers
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Selecting buttons A, B and C.
How do you select buttons A, B and C?Need initial private random seed.
Does not need to be strongly random!
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Final Step
Result may not be uniformly random.
Use known randomness extraction techniques.
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Practicalities
• Actually only need two buttons on each box (using the CHSH inequality).
• Difficult to prevent quantum entanglement quickly degenerating.
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Caveats
• Need to prevent signaling between the boxes.• This “protocol is not yet proven to be universally-
composable against a full quantum adversary”.
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References
•Pironio et al. Random Numbers Certied by Bell's Theorem, Nature v.464, p.1021 (2010).
•J. S. Bell, On the Einstein Podolsky Rosen Paradox, Physics 1, 3, 195-200 (1964)
•A. Einstein, B. Podolsky, and N. Rosen, Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Phys. Rev. 47, 777–780 (1935)
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Schrödinger's Cat