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233-234. Sedgewick & Wayne (2004); Chazelle (2005). Linear-reduces: Cost of reduction is proportional to size of input. Traveling Salesman Problem. Best known algorithm takes exponential time!. P. Problems that can be solved in polynomial time. c.

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233-234

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

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

  3. Traveling Salesman Problem

  4. Best known algorithm takes exponential time!

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

  6. 3-Coloring Not known to be in P

  7. 3-Coloring But is in NP

  8. A polynomial time proof of 3-Coloring

  9. Don’t all problems have polynomial time proofs? No ! Piano mover’s problem Winning strategies

  10. P Problems that can be solved in polynomial time NP 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)

  11. P = NP ?

  12. P Problems that can be solved in polynomial time NP 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 !

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

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

  15. 3-Coloring

  16. 3-Coloring Prover interacts with Verifier

  17. 3-Coloring Prover hides coloring

  18. 3-Coloring Verifier checks an edge at random

  19. 3-Coloring Verifier spots a lie with probability 1/E

  20. 3-Coloring Verifier repeats 100E times

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

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

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

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

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

  26. Step 1: Prover shuffles coloring

  27. Step 2: Prover hides coloring

  28. Step 3: Verifier checks an edge

  29. Step 1: Prover shuffles coloring

  30. Step 2: Prover hides coloring

  31. Step 3: Verifier checks an edge, etc

  32. 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.

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

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

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

  36. compiler

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