1. 233-234233-234 Sedgewick & Wayne (2004); Chazelle (2005) Sedgewick & Wayne (2004); Chazelle (2005)

2. Linear-reduces: Cost of reduction is proportional to size of input

8.  Traveling Salesman Problem

12. Best known algorithm takes exponential time!

17. P NP Problems that can be solved in polynomial time Problems that have polynomial time proofs If input size = N, then time is O(N ) c Suffices to look at Yes/No problems

18. 3-Coloring Not known to be in P

19. 3-Coloring But is in NP

20. A polynomial time proof of 3-Coloring

21. Don’t all problems have polynomial time proofs? Piano mover’s problem Winning strategies

22. P NP Problems that can be solved in polynomial time Problems that have polynomial time proofs (Note that P is symmetric with yes/no but NP is not) COMPOSITE is in NP (easy); so is PRIME (hard)

23.  P = NP ?

24. P NP Problems that can be solved in polynomial time Problems that have polynomial time proofs NP-Complete: Any problem A in NP such that any problem in NP polynomial-reduces to it Over 10,000 known NP-complete problems !

25. FACTORING Given graph G, can it be colored red, white, blue? Given n-bit integer x and k, does x have a factor 1<x<k ? 3-COLOR FACTORING and 3-COLOR are in NP 3-COLOR is NP-complete  3-color efficiently and destroy ALL e-commerce!

26. Zero Knowledge Can I convince you I have a proof without revealing anything about it?

27. 3-Coloring

28. Prover interacts with Verifier

29. 3-Coloring Prover hides coloring

30. 3-Coloring Verifier checks an edge at random

31. 3-Coloring Verifier spots a lie with probability 1/E

32. 3-Coloring Verifier repeats 100E times

33. If Verifier spots no lies, she concludes the graph is 3-colorable Prover fools Verifier with negligible probability

34. Is it Zero-Knowledge? Verifier can color most of the graph!

35. Not Zero-Knowledge! Why do we require the Verifier to check randomly?

36. Repeat 100 E times: 1. Prover: shuffle colors 2. Verifier: Check any edge

37. Random permutation Shuffle colors: what’s that? (6 possibilities)

38. Step 1: Prover shuffles coloring

40. Step 2: Prover hides coloring

41. Step 3: Verifier checks an edge

42. Step 1: Prover shuffles coloring

44. Step 2: Prover hides coloring

45. Step 3: Verifier checks an edge, etc

46. Why is it zero-knowledge? No matter what the Verifier does, she only sees a random pair of colors So, she can simulate the whole protocol by herself – no need for the prover.

47. Every problem in NP has a zero-knowledge proof

48. PCP Can I convince you I have a proof of Riemann’s hypothesis by letting you look at only 2 lines picked at random? (probabilistically checkable proofs) Yes, with probability of error 1/google

49. My proof of RH Slightly longer proof of RH compiler

51. Check two lines If OK, accept proof, else reject The probability of accepting bad proof or rejecting correct proof is < 10 -100