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Elliptic Curves and the State of Survaillence Aleksander Horawa Imperial College London February 21, 2015 Aleksander Horawa Elliptic Curves and the State of Survaillence Reference: Thomas C. Hales, The NSA Back Door to NIST , Notices of the

  1. Elliptic Curves and the State of Survaillence Aleksander Horawa Imperial College London February 21, 2015 Aleksander Horawa Elliptic Curves and the State of Survaillence

  2. Reference: Thomas C. Hales, The NSA Back Door to NIST , Notices of the AMS. Aleksander Horawa Elliptic Curves and the State of Survaillence

  3. image credits: https://wordtothewise.com/ Aleksander Horawa Elliptic Curves and the State of Survaillence

  4. image credits: https://wordtothewise.com/ Aleksander Horawa Elliptic Curves and the State of Survaillence

  5. image credits: https://wordtothewise.com/ Aleksander Horawa Elliptic Curves and the State of Survaillence

  6. image credits: https://wordtothewise.com/ Aleksander Horawa Elliptic Curves and the State of Survaillence

  7. image credits: https://wordtothewise.com/ Aleksander Horawa Elliptic Curves and the State of Survaillence

  8. image credits: https://wordtothewise.com/ Aleksander Horawa Elliptic Curves and the State of Survaillence

  9. One-time pad Alice and Bob both have access to the same (secret) list of random numbers. 5 , 23 , 12 , 15 , 11 , 9 , 3 , 4 , 6 , 24 , 9 , 3 , 6 , 5 , 15 , 7 , 24 , . . . Aleksander Horawa Elliptic Curves and the State of Survaillence

  10. One-time pad Alice and Bob both have access to the same (secret) list of random numbers. 5 , 23 , 12 , 15 , 11 , 9 , 3 , 4 , 6 , 24 , 9 , 3 , 6 , 5 , 15 , 7 , 24 , . . . Alice wants to say ”Hello” to Bob. Aleksander Horawa Elliptic Curves and the State of Survaillence

  11. One-time pad Alice and Bob both have access to the same (secret) list of random numbers. 5 , 23 , 12 , 15 , 11 , 9 , 3 , 4 , 6 , 24 , 9 , 3 , 6 , 5 , 15 , 7 , 24 , . . . Alice wants to say ”Hello” to Bob. Alice h e l l o 7 4 11 11 14 Aleksander Horawa Elliptic Curves and the State of Survaillence

  12. One-time pad Alice and Bob both have access to the same (secret) list of random numbers. 5 , 23 , 12 , 15 , 11 , 9 , 3 , 4 , 6 , 24 , 9 , 3 , 6 , 5 , 15 , 7 , 24 , . . . Alice wants to say ”Hello” to Bob. Alice h e l l o 7 4 11 11 14 + 5 23 12 15 11 Aleksander Horawa Elliptic Curves and the State of Survaillence

  13. One-time pad Alice and Bob both have access to the same (secret) list of random numbers. 5 , 23 , 12 , 15 , 11 , 9 , 3 , 4 , 6 , 24 , 9 , 3 , 6 , 5 , 15 , 7 , 24 , . . . Alice wants to say ”Hello” to Bob. Alice h e l l o 7 4 11 11 14 + 5 23 12 15 11 12 27 23 26 25 Aleksander Horawa Elliptic Curves and the State of Survaillence

  14. One-time pad Alice and Bob both have access to the same (secret) list of random numbers. 5 , 23 , 12 , 15 , 11 , 9 , 3 , 4 , 6 , 24 , 9 , 3 , 6 , 5 , 15 , 7 , 24 , . . . Alice wants to say ”Hello” to Bob. Alice h e l l o 7 4 11 11 14 + 5 23 12 15 11 12 27 23 26 25 (26) 12 1 23 0 25 Aleksander Horawa Elliptic Curves and the State of Survaillence

  15. One-time pad Alice and Bob both have access to the same (secret) list of random numbers. 5 , 23 , 12 , 15 , 11 , 9 , 3 , 4 , 6 , 24 , 9 , 3 , 6 , 5 , 15 , 7 , 24 , . . . Alice wants to say ”Hello” to Bob. Alice h e l l o 7 4 11 11 14 + 5 23 12 15 11 12 27 23 26 25 (26) 12 1 23 0 25 m b x a z Aleksander Horawa Elliptic Curves and the State of Survaillence

  16. One-time pad Alice and Bob both have access to the same (secret) list of random numbers. 5 , 23 , 12 , 15 , 11 , 9 , 3 , 4 , 6 , 24 , 9 , 3 , 6 , 5 , 15 , 7 , 24 , . . . Alice wants to say ”Hello” to Bob. Alice Bob h e l l o m b x a z 7 4 11 11 14 + 5 23 12 15 11 12 27 23 26 25 (26) 12 1 23 0 25 m b x a z Aleksander Horawa Elliptic Curves and the State of Survaillence

  17. One-time pad Alice and Bob both have access to the same (secret) list of random numbers. 5 , 23 , 12 , 15 , 11 , 9 , 3 , 4 , 6 , 24 , 9 , 3 , 6 , 5 , 15 , 7 , 24 , . . . Alice wants to say ”Hello” to Bob. Alice Bob h e l l o m b x a z 7 4 11 11 14 12 1 23 0 25 + 5 23 12 15 11 12 27 23 26 25 (26) 12 1 23 0 25 m b x a z Aleksander Horawa Elliptic Curves and the State of Survaillence

  18. One-time pad Alice and Bob both have access to the same (secret) list of random numbers. 5 , 23 , 12 , 15 , 11 , 9 , 3 , 4 , 6 , 24 , 9 , 3 , 6 , 5 , 15 , 7 , 24 , . . . Alice wants to say ”Hello” to Bob. Alice Bob h e l l o m b x a z 7 4 11 11 14 12 1 23 0 25 + 5 23 12 15 11 − 5 23 12 15 11 12 27 23 26 25 (26) 12 1 23 0 25 m b x a z Aleksander Horawa Elliptic Curves and the State of Survaillence

  19. One-time pad Alice and Bob both have access to the same (secret) list of random numbers. 5 , 23 , 12 , 15 , 11 , 9 , 3 , 4 , 6 , 24 , 9 , 3 , 6 , 5 , 15 , 7 , 24 , . . . Alice wants to say ”Hello” to Bob. Alice Bob h e l l o m b x a z 7 4 11 11 14 12 1 23 0 25 + 5 23 12 15 11 − 5 23 12 15 11 12 27 23 26 25 7 -22 11 - 15 14 (26) 12 1 23 0 25 m b x a z Aleksander Horawa Elliptic Curves and the State of Survaillence

  20. One-time pad Alice and Bob both have access to the same (secret) list of random numbers. 5 , 23 , 12 , 15 , 11 , 9 , 3 , 4 , 6 , 24 , 9 , 3 , 6 , 5 , 15 , 7 , 24 , . . . Alice wants to say ”Hello” to Bob. Alice Bob h e l l o m b x a z 7 4 11 11 14 12 1 23 0 25 + 5 23 12 15 11 − 5 23 12 15 11 12 27 23 26 25 7 -22 11 - 15 14 (26) 12 1 23 0 25 (26) 7 4 11 11 14 m b x a z Aleksander Horawa Elliptic Curves and the State of Survaillence

  21. One-time pad Alice and Bob both have access to the same (secret) list of random numbers. 5 , 23 , 12 , 15 , 11 , 9 , 3 , 4 , 6 , 24 , 9 , 3 , 6 , 5 , 15 , 7 , 24 , . . . Alice wants to say ”Hello” to Bob. Alice Bob h e l l o m b x a z 7 4 11 11 14 12 1 23 0 25 + 5 23 12 15 11 − 5 23 12 15 11 12 27 23 26 25 7 -22 11 - 15 14 (26) 12 1 23 0 25 (26) 7 4 11 11 14 m b x a z h e l l o Aleksander Horawa Elliptic Curves and the State of Survaillence

  22. Random numbers Problem. Need random numbers! How can we generate them? Aleksander Horawa Elliptic Curves and the State of Survaillence

  23. Random numbers Problem. Need random numbers! How can we generate them? Truly random numbers can only come from a physical process. Aleksander Horawa Elliptic Curves and the State of Survaillence

  24. Random numbers Problem. Need random numbers! How can we generate them? Truly random numbers can only come from a physical process. We can generate numbers that appear random from a recipe using a computational device. These are called pseudo-random numbers . Aleksander Horawa Elliptic Curves and the State of Survaillence

  25. Random numbers Problem. Need random numbers! How can we generate them? Truly random numbers can only come from a physical process. We can generate numbers that appear random from a recipe using a computational device. These are called pseudo-random numbers . One method comes from the theory of elliptic curves, which are recently very common in cryptography. Aleksander Horawa Elliptic Curves and the State of Survaillence

  26. Elliptic curves Google Chrome: Key exchange: ECDHE RSA EC = Elliptic Curve Aleksander Horawa Elliptic Curves and the State of Survaillence

  27. Elliptic curves Elliptic curves are a special kind of cubic curves on the plane. Definition An elliptic curve over R is the set of solution ( x , y ) ∈ R 2 of y 2 = x 3 + ax + b for a , b ∈ R such that 27 b 2 + 4 a 3 � = 0, together with a point O called the point at infinity . Aleksander Horawa Elliptic Curves and the State of Survaillence

  28. Elliptic curves Examples y 2 = x 3 − x + 1 y 2 = x 3 − x 3 3 2 2 1 1 − 2 − 1 1 2 − 2 − 1 1 2 − 1 − 1 − 2 − 2 − 3 − 3 Aleksander Horawa Elliptic Curves and the State of Survaillence

  29. Addition on elliptic curves Why are they so useful? You can define addition on them! P Q P + Q Aleksander Horawa Elliptic Curves and the State of Survaillence

  30. Addition on elliptic curves Why are they so useful? You can define addition on them! 2 P P Aleksander Horawa Elliptic Curves and the State of Survaillence

  31. Addition on elliptic curves Why are they so useful? You can define addition on them! P Aleksander Horawa Elliptic Curves and the State of Survaillence

  32. Addition on elliptic curves Problem. The definition is geometric. We need formulas! y = y 2 − y 1 x 2 − x 1 ( x − x 1 ) + y 1 P = ( x 1 , y 1 ) Q = ( x 2 , y 2 ) P + Q = ( x 3 , y 3 ) Aleksander Horawa Elliptic Curves and the State of Survaillence

  33. Addition on elliptic curves Problem. The definition is geometric. We need formulas! y = y 2 − y 1 x 2 − x 1 ( x − x 1 ) + y 1 P = ( x 1 , y 1 ) Q = ( x 2 , y 2 ) � � 2 y 2 − y 1 x 3 = − x 1 − x 2 x 2 − x 1 P + Q = ( x 3 , y 3 ) � � y 2 − y 1 y 3 = − y 1 + ( x 1 − x 3 ) x 2 − x 1 Aleksander Horawa Elliptic Curves and the State of Survaillence

  34. Elliptic curves Computers are good with finite objects. Aleksander Horawa Elliptic Curves and the State of Survaillence

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