Data Matters

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Sedgewick Wayne (2004) Chazelle (2005)
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Linear-reduces Cost of reduction is
proportional to size of input
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• Traveling Salesman Problem

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Best known algorithm takes exponential time!
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P
Problems that can be solved in polynomial time
NP
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)
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• P NP ?

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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 !
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FACTORING
Given n-bit integer x and k, does x have a factor
1ltxltk ?
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!
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Zero Knowledge
Can I convince you I have a proof without
revealing anything about it?
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3-Coloring
Prover interacts with Verifier
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3-Coloring
Prover hides coloring
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3-Coloring
Verifier checks an edge at random
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3-Coloring
Verifier spots a lie with probability 1/E
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3-Coloring
Verifier repeats 100E times
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If Verifier spots no lie, she concludes the graph
is 3-colorable
Prover fools Verifier with negligible probability
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Is it Zero-Knowledge?
Verifier can color most of the graph!
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Not Zero-Knowledge!
Why do we require the Verifier to check randomly?
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Repeat 100 E times
1. Prover shuffle colors
2. Verifier Check any edge
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Shuffle colors whats that?
Random permutation
(6 possibilities)
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To summarize
Step 1 Prover shuffles coloring
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Step 2 Prover hides coloring
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Step 3 Verifier checks an edge
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Step 1 Prover shuffles coloring
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Step 2 Prover hides coloring
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Step 3 Verifier checks an edge, etc
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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.
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PCP
(probabilistically checkable proofs)
Can I convince you I have a proof of Riemanns
hypothesis by letting you look at only 2 lines
picked at random?
Yes, with probability of error 1/google