127 lines
3.0 KiB
TeX
Executable File
127 lines
3.0 KiB
TeX
Executable File
\section{Diffie-Hellman Key Exchange}
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One problem we encounter in computer science is \textit{secure key exchange}: How can two parties (usually called Alice and Bob) agree on a \say{key} without revealing anything to an eavesdropper (Eve)?
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\begin{center}
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\begin{tikzpicture}
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\node (A) at (0, 0) {Alice};
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\node (B) at (4, 0) {Bob};
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\node (E) at (2, -1) {Eve};
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\draw[-]
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(A) edge (B)
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(E) edge (2, 0)
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;
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\end{tikzpicture}
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\end{center}
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A simple mathematical solution to the key exchange problem is the \textit{Diffie-Hellman key exchange algorithm}, detailed below.
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\vspace{1mm}
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Values that are \textit{public} are known to everyone. Values that are sent are also known to everyone: we assume that everyone can see what Alice and Bob send to each other.
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Eve can read all public values, but she cannot change them in any way.
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\begin{center}
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\begin{tikzpicture}[scale = 0.5]
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\def\bx{18}
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\def\ex{13}
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\node[anchor = center] at (\ex, 7.5) {\textbf{Setup}};
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\draw[-] (\ex-4.5, 7) -- (\ex+4.5, 7);
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\node[anchor = west] at (\ex-4, 6) {Let $p$ be a prime number};
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\node[anchor = west] at (\ex-4, 5) {Let $g$ be a generator in $\mathbb{Z}_p^\times$};
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\node[anchor = west] at (\ex-4, 4) {Both $g$ and $p$ are public.};
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\node[anchor = center] at (4, 1.5) {\textbf{Alice}};
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\draw[-] (-0.5, 1) -- (8.5, 1);
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\node[anchor = west] at (0, 0) {Pick a random $a \in \mathbb{Z}_p^\times$};
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\node[anchor = west] at (0, -1) {Set $A = g^a$};
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\node[anchor = west] at (0, -3) {Publish $A$};
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\draw[->] (6, -3) -- (\ex - 1, -3);
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\node[anchor = west] at (0, -5) {\color{gray} Compute ...};
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\node[anchor = center] at (\bx+4, 1.5) {\textbf{Bob}};
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\draw[-] (\bx-0.5, 1) -- (\bx+8.5, 1);
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\node[anchor = west] at (\bx, 0) {Pick a random $b \in \mathbb{Z}_p^\times$};
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\node[anchor = west] at (\bx, -1) {Set $B = g^b$};
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\node[anchor = west] at (\bx, -4) {Publish $B$};
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\draw[->] (\bx - 1, -4) -- (\ex+1, -4);
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\node[anchor = west] at (\bx, -5) {\color{gray} Compute ...};
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\node[anchor = center] at (\ex, 1.5) {\textbf{Public}};
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\draw[-] (\ex-2, 1) -- (\ex+2, 1);
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\node[anchor = center] at (\ex, 0) {$p, g$};
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\node[fill=white, anchor = center] at (\ex, -3) {$A$};
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\node[fill=white, anchor = center] at (\ex, -4) {$B$};
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\end{tikzpicture}
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\end{center}
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\problem{}
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Complete the algorithm. What should Alice and Bob compute? \par
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What is their shared secret?
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\vfill
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\problem{}
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Let $p = 11$, $g = 2$, $a = 9$, and $b = 4$. \par
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Run the algorithm. What is the resulting shared secret?
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\begin{solution}
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$g^b = 5$\par
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$g^a = 6$\par
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$g^{ab} = g^{ba} = 9$
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\end{solution}
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\vfill
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\pagebreak
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\problem{}
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Is the Diffie-Hellman key exchange algorithm secure? What information does Eve have? \par
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What does Eve need to do to find the value Alice and Bob agreed on?
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\vfill
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\problem{}
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Now, say Eve can change information in transit. \par
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That is, she can pretend to be Alice to send information to Bob. \par
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How can she break this system? \par
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\note[Note]{This is called a \textit{man-in-the-middle} attack.}
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\vfill
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\pagebreak
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