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 Suffices to look at Yes/No problems (Note that P is symmetric with yes/no but NP is not) COMPOSITE is in NP (easy); so is PRIME (hard)

18.  P = NP ?

19. 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 !

20. 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!

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

22. 3-Coloring Prover interacts with Verifier

23. 3-Coloring Prover hides coloring

24. 3-Coloring Verifier checks an edge at random

25. 3-Coloring Verifier spots a lie with probability 1/E

26. 3-Coloring Verifier repeats 100E times

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

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

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

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

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

33. To summarize Step 1: Prover shuffles coloring

34. Step 2: Prover hides coloring

35. Step 3: Verifier checks an edge

36. Step 1: Prover shuffles coloring

37. Step 2: Prover hides coloring

38. Step 3: Verifier checks an edge, etc

39. 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.

40. 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