Categories: Cryptography.

In cryptography, the **Diffie-Hellman key exchange** is a method for two parties to securely agree on an encryption key, when they can only communicate over an insecure channel.

The fundamental assumption of the Diffie-Hellman scheme, upon which its security rests, is that the following function \(f(n)\) is a **trapdoor function**, which means that calculating \(f\) is easy, but its inverse \(f^{-1}\) is extremely hard to find:

\[\begin{aligned} f(n) = g^n \bmod p \end{aligned}\]

Where \(n\) is a natural number, and \(p\) is a prime. The natural number \(g\) is a so-called *primitive root modulo* \(p\). Importantly, \(g\) and \(p\) have been specifically chosen such that \(f(n)\) can take any value in \(\{1, ..., p \!-\! 1\}\) for \(n\) in \(\{0, ..., p \!-\! 2\}\). The trapdoor assumption is that, given \(g\), \(p\) and \(f(n)\), there is no efficient algorithm to recover \(n\).

Suppose that Alice and Bob want to exchange encrypted data in the future, so they need to agree on an encryption key to use. However, they can only exchange messages with each other over an insecure channel, which is being eavesdropped.

After they publicly agree on the values of \(g\) and \(p\), Alice and Bob each choose a secret number from \(\{0, ..., p \!-\! 2\}\), respectively \(a\) and \(b\), and then privately calculate \(A\) and \(B\) as follows:

\[\begin{aligned} A = g^a \bmod p \qquad \quad B = g^b \bmod p \end{aligned}\]

Finally, they transmit these numbers \(A\) and \(B\) to each other over the insecure connection, and then each side calculates \(k\), which is the desired secret key:

\[\begin{aligned} \boxed{ k = A^b \bmod p = B^a \bmod p = g^{ab} \bmod p } \end{aligned}\]

The point is that \(k\) includes both \(a\) *and* \(b\), but each side only needs to know *either* \(a\) *or* \(b\). And, due to the trapdoor assumption, the eavesdropper knows \(A\) and \(B\), but cannot recover \(a\) or \(b\).

This assumption is just that: an assumption. So far, nobody has been able to prove or disprove it for classical computation. However, for quantum computers, it has already been *dis*proven! In this case, another method must be used, for example the BB84 protocol.

- J.B. Brask,
*Quantum information: lecture notes*, 2021, unpublished.

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