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Minimizing the number of keys for secure communication in a network

Minimizing the number of keys for secure communication in a network. By Niels Duif. Remarks. Ask questions Proof by example. Contents. Introduction Splitting a message Constructions Combining constructions Conclusions. Introduction. Network Symmetric cryptography. Introduction.

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Minimizing the number of keys for secure communication in a network

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  1. Minimizing the number of keys for secure communication in a network By Niels Duif

  2. Remarks Ask questions Proof by example / Department of Mathematics and Computer Science

  3. Contents Introduction Splitting a message Constructions Combining constructions Conclusions / Department of Mathematics and Computer Science

  4. Introduction Network Symmetric cryptography / Department of Mathematics and Computer Science

  5. Introduction Chris Eve Alice common key: kAB Bob Network Symmetric cryptography / Department of Mathematics and Computer Science

  6. Introduction Chris Eve Hi! Alice common key: kAB Bob Network Symmetric cryptography / Department of Mathematics and Computer Science

  7. Introduction Chris Eve z$# Alice common key: kAB Bob Network Symmetric cryptography / Department of Mathematics and Computer Science

  8. Introduction Chris Eve z$# Alice common key: kAB Bob Network Symmetric cryptography / Department of Mathematics and Computer Science

  9. Introduction Chris Eve Alice z$# common key: kAB Bob Network Symmetric cryptography / Department of Mathematics and Computer Science

  10. Introduction Chris Eve Alice Hi! common key: kAB Bob Network Symmetric cryptography / Department of Mathematics and Computer Science

  11. Introduction Chris Eve ? Alice Hi! common key: kAB Bob Network Symmetric cryptography / Department of Mathematics and Computer Science

  12. Introduction common key: kCE Chris Eve Alice common key: kAB common key: kBC Bob Network Symmetric cryptography / Department of Mathematics and Computer Science

  13. Communication graph Alice Bob Chris Eve / Department of Mathematics and Computer Science

  14. Communication graph Alice Bob Chris Eve Persons are represented by nodes Nodes that have one or more keys in common are connected by a line / Department of Mathematics and Computer Science

  15. Communication graph kAB kBC kCE Alice Bob Chris Eve Persons are represented by nodes Nodes that have one or more keys in common are connected by a line The lines are labelled with the common keys / Department of Mathematics and Computer Science

  16. Communication graph 1 2 3 A B C E Persons are represented by nodes Nodes that have one or more keys in common are connected by a line The lines are labelled with the common keys / Department of Mathematics and Computer Science

  17. Communication graph A B 1 5 4 2 3 C D Nodes may have more than one key in common / Department of Mathematics and Computer Science

  18. Communication graph A (1,2,5) B (1,4,5) 1 5 4 2 3 C (2,3) D (3,4) Nodes may have more than one key in common / Department of Mathematics and Computer Science

  19. How to assign keys? A E B C D / Department of Mathematics and Computer Science

  20. How to assign keys? A E B C D Give every pair of nodes a different key / Department of Mathematics and Computer Science

  21. How to assign keys? A E B C D Give every pair of nodes a different key / Department of Mathematics and Computer Science

  22. How to assign keys? A E B C D • Give every pair of nodes a different key • This requires keys for n nodes / Department of Mathematics and Computer Science

  23. Using fewer keys • Is secure communication possible with fewer keys? • Yes, assuming that some nodes may be trusted • Assumption: at most t nodes cannot be trusted • Aim: minimize the total number of keys, c / Department of Mathematics and Computer Science

  24. Splitting a message / Department of Mathematics and Computer Science

  25. Splitting a message A (1,2) B (1,4) 1 4 2 3 C (2,3) D (3,4) Split a message and send it through different paths Example: communication from A to D / Department of Mathematics and Computer Science

  26. Splitting a message A (1,2) B (1,4) 1 part 1 4 2 3 C (2,3) D (3,4) Split a message and send it through different paths Example: communication from A to D / Department of Mathematics and Computer Science

  27. Splitting a message A (1,2) B (1,4) 1 part 1 4 2 part 2 3 C (2,3) D (3,4) Split a message and send it through different paths Example: communication from A to D / Department of Mathematics and Computer Science

  28. Splitting a message Determine random shares: M1, M2, ... , Ms-1 Use bitwise addition mod 2: “ ”, or “XOR” Ms = M1 M2 ... Ms-1 M M1 M2 ... Ms-1 Ms = M / Department of Mathematics and Computer Science

  29. Splitting a message • Example: the message is ‘Hi!’ M0 = 01001000 01101001 00100001 M1 = 11101101 11101111 10010001 M2 = 10100101 10000110 10110000 / Department of Mathematics and Computer Science

  30. Splitting a message • Example: the message is ‘Hi!’ M0 = 01001000 01101001 00100001 M1 = 11101101 11101111 10010001 M2 = 10100101 10000110 10110000 M1 = 11101101 11101111 10010001 M2 = 10100101 10000110 10110000 / Department of Mathematics and Computer Science

  31. Splitting a message • Example: the message is ‘Hi!’ M0 = 01001000 01101001 00100001 M1 = 11101101 11101111 10010001 M2 = 10100101 10000110 10110000 M1 = 11101101 11101111 10010001 M2 = 10100101 10000110 10110000 M0 = 01001000 01101001 00100001 / Department of Mathematics and Computer Science

  32. Splitting a message • Example: the message is ‘Hi!’ M0 = 01001000 01101001 00100001 M1 = 11101101 11101111 10010001 M2 = 10100101 10000110 10110000 M1 = 11101101 11101111 10010001 M2 = 10100101 10000110 10110000 M0 = 01001000 01101001 00100001 / Department of Mathematics and Computer Science

  33. Sending a message A (1,2) B (1,4) 1 M1 4 2 3 C (2,3) D (3,4) / Department of Mathematics and Computer Science

  34. Sending a message A (1,2) B (1,4) 1 M1 4 2 M2 3 C (2,3) D (3,4) / Department of Mathematics and Computer Science

  35. Retrieving a message The message is retrieved as the XOR of all shares: M = M1 M2 ... Ms All shares are needed to retreive the message / Department of Mathematics and Computer Science

  36. Constructions for t=1 One corrupt node The total number of keys is c How large can n be? / Department of Mathematics and Computer Science

  37. Sperner’s theorem n is at most This uses all possible key sets of size / Department of Mathematics and Computer Science

  38. Example / Department of Mathematics and Computer Science

  39. Example Communication from A to F / Department of Mathematics and Computer Science

  40. Example C (1,4) B (1,3) E(2,4) 4 1 D (2,3) 3 2 F (3,4) A (1,2) Communication from A to F / Department of Mathematics and Computer Science

  41. Example C (1,4) B (1,3) E(2,4) 4 1 D (2,3) 3 2 F (3,4) A (1,2) Communication from A to F / Department of Mathematics and Computer Science

  42. Example C (1,4) B (1,3) E(2,4) 4 1 D (2,3) 3 2 F (3,4) A (1,2) • Communication from A to F • Use all possible combinations of A’s and F’s keys / Department of Mathematics and Computer Science

  43. Example C (1,4) B (1,3) E(2,4) 4 1 D (2,3) 3 2 F (3,4) A (1,2) • Communication from A to F • Use all possible combinations of A’s and F’s keys / Department of Mathematics and Computer Science

  44. Example C (1,4) B (1,3) E(2,4) 4 1 D (2,3) 3 2 F (3,4) A (1,2) • Communication from A to F • Use all possible combinations of A’s and F’s keys / Department of Mathematics and Computer Science

  45. Example C (1,4) B (1,3) E(2,4) 4 1 D (2,3) 3 2 F (3,4) A (1,2) • Communication from A to F • Use all possible combinations of A’s and F’s keys / Department of Mathematics and Computer Science

  46. Example C (1,4) B (1,3) E(2,4) 4 1 D (2,3) 3 2 F (3,4) A (1,2) • Communication from A to F • Use all possible combinations of A’s and F’s keys / Department of Mathematics and Computer Science

  47. Example C (1,4) B (1,3) E(2,4) 4 1 D (2,3) 3 2 F (3,4) A (1,2) • Communication from A to F • Use all combinations of their keys / Department of Mathematics and Computer Science

  48. Eavesdrop The following keys are needed: / Department of Mathematics and Computer Science

  49. Eavesdrop • Key 1 or Key 3 The following keys are needed: / Department of Mathematics and Computer Science

  50. Eavesdrop • Key 1 or Key 3 • Key 1 or Key 4 The following keys are needed: / Department of Mathematics and Computer Science

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