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

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Presentation transcript:

Sedgewick & Wayne (2004); Chazelle (2005) 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 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)

 P = NP ?

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 !

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!

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

3-Coloring Prover interacts with Verifier

3-Coloring Prover hides coloring

3-Coloring Verifier checks an edge at random

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

3-Coloring Verifier repeats 100E times

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

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

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

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

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

To summarize Step 1: Prover shuffles coloring

Step 2: Prover hides coloring

Step 3: Verifier checks an edge

Step 1: Prover shuffles coloring

Step 2: Prover hides coloring

Step 3: Verifier checks an edge, etc

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.

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