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\section*{Handout 6 (Zero-Knowledge Proofs)}

Zero-knowledge proofs (short ZKP) solve a paradoxical puzzle:

How to convince somebody else that one knows a secret without,

revealing what the secret is? This sounds like a problem the

Mad Hatter from Alice in Wonderland would occupy himself with,

but actually there some serious and not so serious

applications of it. For example, if you solve crosswords with

your friend, say Bob, you might want to convince him that you

found a solution for one question, but of course you do not

want to reveal the solution, as this might give Bob an

advantage somewhere else in the crossword. So how to convince

Bob that you know the answer (secret)? One way would be to

come up with the following protocol: suppose the answer is

\textit{folio}. Then look up the definition of that word in a

dictionary. Say you find

\begin{quote}

``an \textit{individual} leaf of paper or parchment, either

loose as one of a series or forming part of a bound volume,

which is numbered on the recto or front side only.''

\end{quote}

\noindent

Take the first non-article word in this definition,

in this case \textit{individual}, and look up the definition

of this word, say

\begin{quote}

``a single \textit{human} being as distinct from a group''

\end{quote}

\noindent 

In this definition take the second non-article word, that

is \textit{human}, and again look up the definition of this 

word. This will yield

\begin{quote}

``relating to \textit{or} characteristic of humankind''

\end{quote}

\noindent 

You could now go on to look up the definition of the third

non-article in this definition and so on. But let us assume

you agreed with Bob to stop after three iterations with the

third non-article word in the last definition, that is

\textbf{or}. Now, instead of sending to Bob the solution

\textit{folio}, you send to him \textit{or}. How can

Bob verify that you know the solution? Well, once he solved

it himself, he can use the dictionary and follow the same

``trail'' as you did. It the final word agrees with what 

you send him, he must infer you know the solution earlier

than him. This protocol works like a one-way hash function 

because \textit{or} does not give any hint as to what was

the first word was. I leave you to think why this

protocol avoids article words. 

After Bob found his solution and verified that according to

the protocol it ``maps'' also to \textit{or}, can he be

entirely sure no cheating is going on. Not with 100\%

certainty. It could have been entirely possible that

he was given \textit{or} as the word but it derived

from an entirely different word---this seems very unlikely,

but is at least theoretical a possibility. Protocols

based on zero-knowledge proofs will produce a similar

result---they give an answer that might be erroneous 

in a very small number of cases. The point is to itterate

them long enough so that the theoretical possibility of

cheating is negligibly small. 

By the way, the authors of the paper ``Dismantling Megamos

Crypto: Wirelessly Lockpicking a Vehicle Immobilizer'' who

were barred from publishing their results used also a hash to

prove they did the work and (presumably) managed to get into

cars without a key. See Figure~\ref{paper}. This is very 

similar to the method about crosswords. They like to prove

that they did the work, but not giving out the ``solution''. 

But this also shows what the problem with such methods

are: yes, we can hide the secret temporarily, but if

somebody else wants to verify it, then the secret has 

to be made public. Bob needs to know that \textit{folio}

is the solution before he can verify the claim that somebody

else had the solution first. Similarly with the paper:

we need to wait until the authors are finally allowed to 

publish their finding in order to verify the hash. This

might happen at some point, but equally it might never

happen (what for example happens if the authors loose

their copy of the paper because of a disk failure?).

Zero-knowledge proofs, in contrast, can be immediately

be checked, even if the secret is not public yet.

\begin{figure}

\begin{center}

\fbox{\includegraphics[scale=0.4]{../pics/Dismantling_Megamos_Crypto.png}}

\end{center}

\caption{The authors of this paper used a hash in order to prove

later that they have managed to break into cars.\label{paper}}

\end{figure}

\subsubsection*{ZKP: An Illustrative Example}

The idea behind zero-knowledge proofs is not very obvious and

will surely take some time for you to digest. Therefore

lets start with an illustrative example. The example will

not be perfect, but hopefully explain the gist of the ideas.

The example is called Alibaba's cave:

\begin{center}

\begin{tabular}{c@{\hspace{8mm}}c@{\hspace{8mm}}c}

\includegraphics[scale=0.1]{../pics/alibaba1.png} &

\includegraphics[scale=0.1]{../pics/alibaba2.png} &

\includegraphics[scale=0.1]{../pics/alibaba3.png} \\

Step 1 & Step 2 & Step 3

\end{tabular}

\end{center}

\noindent Lets have a look at the picture in Step 1:

The cave is a loop where at the end is a magic wand.

The point of the magic wand is that Alice knows the

secret word for how to open it. She wants to keep here

word secret, but wants to convince Bob that she knows it.

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