My understanding is that there is a difference between the concept of a Zero-Knowledge Proof (ZKP), and then the applications that such a thing is possible.

In the example given, I can prove that N is composite without revealing anything (well, almost anything) about the factors. But in practice we want to use a ZKP to show that I have specific knowledge without revealing the knowledge itself.

For example:

You can give me a graph, and I can claim that I can three-colour it. You may doubt this, but there is a process by which I can ... to any desired level of confidence ... demonstrate that I have a colouring, without revealing what the colouring is. I colour the vertices RGB, map those colours randomly to ABC, and cover all the vertices. You choose any edge, and I reveal the "colours" (from ABC) of the endpoints. If I really can colour the graph then I will always be able to reveal two different colours. If I can't colour the graph then as we do this more and more, eventually I will fail.

So you are right, but the message of the post is, I think, still useful and relevant.

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mathgradthrow 3 hours ago | parent [-]

can you explain this a little better?

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ColinWright 3 hours ago | parent | next [-]

I can certainly explain it more, a question of "better" is debatable!

Here's the process:

(A) You give me a graph to 3-colour;

(B) I claim I can 3-colour it;

(D) I colour it with colours ABC and cover the vertices;

(E) You point at an edge;

(F) I reveal the colours of the vertices at the ends of the edge;

(G) If I have coloured the graph then the colours revealed will always be different;

(H) We repeat this process with a permutation of the colours between each trial;

(I) If I'm lying then eventually you'll pick an edge where either the vertices are not coloured, or the have the same colour.

(J) This process reveals nothing about the colouring, but proves (to some level of confidence) that I'm telling the truth.

So ... what's unclear?

Instructions on how to email me are in my profile if you prefer ...

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kadoban 2 hours ago | parent [-]

How do I know/prove that you're not just saying any random two colors for whichever edge I choose?

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ColinWright 2 hours ago | parent [-]

The version I'm describing has it physically sitting in front of you at the time, so you can see that the colours haven't been changed "on the fly" after you pick an edge. In this version:

(A) I colour it;

(B) I cover the vertices so you can't see any of them, but I can no longer change them;

Converting this to a digital version requires further work ... my intent here was to explain the underlying idea that I can prove (to some degree of confidence) that I have a colouring without revealing anything about it.

So just off the top of my head, for example, I can, for each vertex, create a completely random string that starts with "R", "G", or "B" depending on the colour of the vertex. Then I hash each of those, and send you all of them. You choose an edge and send me back the two hashes for the endpoints, and I provide the associated random strings so you can check that the hashes match.

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lordnacho 35 minutes ago | parent [-]

This reminds me of the "Where's Waldo (Wally in UK)" example:

You can prove that you found Wally with a large piece of paper with a hole in it. You move the hole over Wally, and the person you're sitting with can see you found it, but he's no wiser about where.

	▲	fragmede 7 minutes ago \| parent [-]
		https://youtu.be/5qzNe1hk0oY for a video if you can't picture that.

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fragmede 3 hours ago | parent | prev [-]

The key insight is that Colin can show you a red-green-blue coloring of the graph, and flip the whole graph secretly, so it's blue-red-green instead when you look at an individual section, but really the graph is yellow-pink-orange colored. Even after showing you all the intersections of the graph individually in the red green blue coloring to satisfy that he can 3-color it, you still have no idea what is yellow pink or orange on his copy of the graph.