The document summarizes a method for generating verifiably random numbers using the principles of quantum entanglement and Bell's theorem. It involves measuring the spin properties of entangled electron pairs from different angles and showing their results violate Bell's inequality, proving the results are genuinely random. The technique could be used to produce cryptographically secure random numbers for applications like key generation, with some practical challenges around maintaining entanglement and preventing signaling between the measurement devices.

1. Random Numbers Certified by Bell's Theoremby Pironio and 10 other authorsfrom 6 different universities and institutions. A brief introduction by David Kemp Licensed under a Creative Commons Attribution 3.0 License http://creativecommons.org/licenses/by/3.0/

2. 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.

4. Proved “reality not separable from observation”.

6. Only one blue-eyed female.

7. How many (max) blue-eyed right-handers?

8. At most one of them can be male.

9. At most one of them can be female.

11. In Probabilistic Terms Pr(blue-eyed right-hander) ≤ Pr(blue-eyed female) + Pr(right-handed male)

13. Are you a blue-eyed right-hander?

14. Are you a blue-eyed female?

16. Are you a blue-eyed right-hander?

17. Are you a blue-eyed female?

18. 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!

20. No communication between the rooms.

21. Ask every person only one of these questions:

22. Are you blue-eyed?

23. Are you left-handed?

24. Are you female?

25. Sometimes ask both twins the same question.

27. A', B' and C' are properties of twin,

28. Pr(x & not(x')) = 0.THEN Pr(A & not(B')) + Pr(B & not(C')) ≥ Pr(A & not(C'))

29. A B C A B C Pairs of Magic Boxes Three buttons and two lights on each box

30. A B C A B C Magic Box Pairs Can only press one button on each box.

31. A B C A B C Magic Box Properties Whenever same button is pressed on both boxes: Same light flashes

33. Pr(B:red & C:yellow) ≈ 7.5 %

34. Pr(A:red & C:yellow) ≈ 25%Pr(A:red & B:yellow) + Pr(B:red & C:yellow)< Pr(A:red & C:yellow)

35. But surely that is impossible!!! Image by Wild Guru Larry. Some rights reserved. http://www.flickr.com/photos/wentzelepsy/4435803492/

36. Relies on Quantum Entanglement

38. Spin-up/down instead of clockwise/anti-clockwise

39. Result seems to be completely random.

40. Can measure from different angles.

41. Eg. Vertically, horizontally, and 45º to horizon.

42. Measurement disturbs the electron.

44. Entangled Electrons Measure entangled electrons from same direction: 100% Correlation (Press same button on both boxes and lights agree)

46. A:yellow& C:yellow 25%

47. A:red & C:yellow 25%

48. A:yellow & C:red 25%Pr(A:red & B:yellow) + Pr(B:red & C:yellow) < Pr(A:red & C:yellow)

50. A:yellow& B:yellow ≈ 42.5%

51. A:red & B:yellow ≈ 7.5%

52. A:yellow & B:red ≈ 7.5%Pr(A:red & B:yellow) + Pr(B:red & C:yellow) < Pr(A:red & C:yellow)

54. B:yellow& C:yellow ≈ 42.5%

55. B:red & C:yellow ≈ 7.5%

56. B:yellow & C:red ≈ 7.5%Pr(A:red & B:yellow) + Pr(B:red & C:yellow) < Pr(A:red & C:yellow)

57. Quantum Entanglement is the only known process thatcan violate Bell’s Inequality.

58. Verifiably Random We can use these magic boxes to generate verifiably random numbers

59. 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!

60. Final Step Result may not be uniformly random. Use known randomness extraction techniques.

64. J. S. Bell, On the Einstein Podolsky Rosen Paradox, Physics 1, 3, 195-200 (1964)