Books / Crypto 101 / Chapter 4

Key exchange

Key exchange protocols attempt to solve a problem that, at first glance, seems impossible. Alice and Bob, who’ve never met before, have to agree on a secret value. The channel they use to communicate is insecure: we’re assuming that everything they send across the channel is being eavesdropped on.

We’ll demonstrate such a protocol here. Alice and Bob will end up having a shared secret, only communicating over the insecure channel. Despite Eve having literally all of the information Alice and Bob send to each other, she can’t use any of that information to figure out their shared secret.

That protocol is called Diffie-Hellman, named after Whitfield Diffie and Martin Hellman, the two cryptographic pioneers who discovered it. They suggested calling the protocol Diffie-Hellman-Merkle key exchange, to honor the contributions of Ralph Merkle. While his contributions certainly deserve honoring, that term hasn’t really caught on. For the benefit of the reader we’ll use the more common term.

Practical implementations of Diffie-Hellman rely on mathematical problems that are believed to be very complex to solve in the “wrong” direction, but easy to compute in the “right” direction. Understanding the mathematical implementation isn’t necessary to understand the principle behind the protocol. Most people also find it a lot easier to understand without the mathematical complexity. So, we’ll explain Diffie-Hellman in the abstract first, without any mathematical constructs. Afterwards, we’ll look at two practical implementations.

Abstract Diffie-Hellman

In order to describe Diffie-Hellman, we’ll use an analogy based on mixing colors. We can mix colors according to the following rules:

  • It’s very easy to mix two colors into a third color.
  • Mixing two or more colors in different order results in the same color.
  • Mixing colors is one-way. It’s impossible to determine if, let alone which, multiple colors were used to produce a given color. Even if you know it was mixed, and even if you know some of the colors used to produce it, you have no idea what the remaining color(s) were.

We’ll demonstrate that with a mixing function like this one, we can produce a secret color only known by Alice and Bob. Later, we’ll simply have to describe the concrete implementation of those functions to get a concrete key exchange scheme.

To illustrate why this remains secure in the face of eavesdroppers, we’ll walk through an entire exchange with Eve, the eavesdropper, in the middle. Eve is listening to all of the messages sent across the network. We’ll keep track of everything she knows and what she can compute, and end up seeing why Eve can’t compute Alice and Bob’s shared secret.

To start the protocol, Alice and Bob have to agree on a base color. They can communicate that across the network: it’s okay if Eve intercepts the message and finds out what the color is. Typically, this base color is a fixed part of the protocol; Alice and Bob don’t need to communicate it. After this step, Alice, Bob and Eve all have the same information: the base color.

/img/books/crypto101/alice-bob-eve.svg

Alice and Bob both pick a random color, and they mix it with the base color.

/img/books/crypto101/alice-bob-secret.svg

At the end of this step, Alice and Bob know their respective secret color, the mix of the secret color and the base color, and the base color itself. Everyone, including Eve, knows the base color.

/img/books/crypto101/alice-bob-eve-secret.svg

Then, Alice and Bob both send their mixed colors over the network. Eve sees both mixed colors, but she can’t figure out what either of Alice and Bob’s secret colors are. Even though she knows the base, she can’t “un-mix” the colors sent over the network.1

/img/books/crypto101/mixed-secret.svg

At the end of this step, Alice and Bob know the base, their respective secrets, their respective mixed colors, and each other’s mixed colors. Eve knows the base color and both mixed colors.

/img/books/crypto101/alice-bob-eve-mixed.svg

Once Alice and Bob receive each other’s mixed color, they add their own secret color to it. Since the order of the mixing doesn’t matter, they’ll both end up with the same secret.

/img/books/crypto101/alice-bob-shared-mixed.svg

Eve can’t perform that computation. She could finish the computation with either Alice or Bob’s secret color, since she has both mixed colors, but she has neither of those secret colors. She can also try to mix the two mixed colors, which would have both Alice and Bob’s secret colors mixed into them. However, that would have the base color in it twice, resulting in a different color than the shared secret color that Alice and Bob computed, which only has the base color in it once.

Diffie-Hellman with discrete logarithms

This section describes a practical implementation of the Diffie-Hellman algorithm, based on the discrete logarithm problem. It is intended to provide some mathematical background, and requires modular arithmetic to understand.

Discrete log Diffie-Hellman is based on the idea that computing \(y\) in the following equation is easy (at least for a computer):

\[y \equiv g^x \pmod{p}\]

However, computing \(x\) given \(y\), \(g\) and \(p\) is believed to be very hard. This is called the discrete logarithm problem, because a similar operation without the modular arithmetic is called a logarithm.

This is just a concrete implementation of the abstract Diffie-Hellman process we discussed earlier. The common base color is a large prime \(p\) and the base \(g\). The “color mixing” operation is the equation given above, where \(x\) is the input value and \(y\) is the resulting mixed value.

When Alice or Bob select their random numbers \(r_A\) and \(r_B\), they mix them with the base to produce the mixed numbers \(m_A\) and \(m_B\):

\[m_A \equiv g^{r_A} \pmod{p}\] \[m_B \equiv g^{r_B} \pmod{p}\]

These numbers are sent across the network where Eve can see them. The premise of the discrete logarithm problem is that it is okay to do so, because figuring out \(r\) in \(m \equiv g^r \pmod{p}\) is supposedly very hard.

Once Alice and Bob have each other’s mixed numbers, they add their own secret number to it. For example, Bob would compute:

\[s \equiv (g^{r_A})^{r_B} \pmod{p}\]

While Alice’s computation looks different, they get the same result, because \((g^{r_A})^{r_B} \equiv (g^{r_B})^{r_A} \pmod{p}\). This is the shared secret.

Because Eve doesn’t have \(r_A\) or \(r_B\), she can not perform the equivalent computation: she only has the base number \(g\) and mixed numbers \(m_A \equiv g^{r_A} \pmod{p}\) and \(m_B \equiv g^{r_B} \pmod{p}\) , which are useless to her. She needs either \(r_A\) or \(r_B\) (or both) to make the computation Alice and Bob do.

Diffie-Hellman with elliptic curves

This section describes a practical implementation of the Diffie-Hellman algorithm, based on the elliptic curve discrete logarithm problem. It is intended to provide some mathematical background, and requires a (very basic) understanding of the mathematics behind elliptic curve cryptography. If you are unfamiliar with elliptic curves, you can either skip this chapter.

One of the benefits of the elliptic curve Diffie-Hellman variant is that the required key size is much, much smaller than the variant based on the discrete log problem. This is because the fastest algorithms for breaking the discrete log problem have a larger asymptotic complexity than their elliptic curve variants. For example, the number field sieve for discrete logarithms, a state of the art algorithm for attacking discrete logarithm-based Diffie-Hellman, has time complexity:

\[L\left[1/3,\sqrt[3]{64/9}\right]\]

Which is more than polynomial (but less than exponential) in the number of digits. On the other hand, the fastest algorithms that could be used to break the elliptic curve discrete log problem all have complexity:

\[L\left[1, 1/2\right] = O(\sqrt{n})\]

Relatively speaking, that means that it’s much harder to solve the elliptic curve problem than it is to solve the regular discrete log problem, using state of the art algorithms for both. The flip side of that is that for equivalent security levels, the elliptic curve algorithm needs much smaller key sizes:

Security level in bits

Discrete log key bits

Elliptic curve key bits

56

512

112

80

1024

160

112

2048

224

128

3072

256

256

15360

512

Remaining problems

Using Diffie-Hellman, we can agree on shared secrets across an insecure Internet, safe from eavesdroppers. However, while an attacker may not be able to simply get the secret from eavesdropping, an active attacker can still break the system. If such an attacker, usually called Mallory, is in between Alice and Bob, she can still perform the Diffie-Hellman protocol twice: once with Alice, where Mallory pretends to be Bob, and once with Bob, where Mallory pretends to be Alice.

/img/books/crypto101/MITM.svg

There are two shared secrets here: one between Alice and Mallory, and one between Mallory and Bob. The attacker (Mallory) can then simply take all the messages they get from one person and send them to the other, they can look at the plaintext messages, remove messages, and they can also modify them in any way they choose.

To make matters worse, even if one of the two participants was somehow aware that this was going on, they would have no way to get the other party to believe them. After all: Mallory performed the successful Diffie-Hellman exchange with the unwitting victim, she has all the correct shared secrets. Bob has no shared secrets with Alice, just with Mallory; there’s no way for him to prove that he’s the legitimate participant. As far as Alice can tell, Bob just chose a few random numbers. There’s no way to link any key that Bob has with any key that Alice has.

Attacks like these are called MITM attacks, because the attacker (Mallory) is in between the two peers (Alice and Bob). Given that the network infrastructure that we typically use to send messages is run by many different operators, this kind of attack scenario is very realistic, and a secure cryptosystem will have to address them somehow.

While the Diffie-Hellman protocol successfully produced a shared secret between two peers, there are clearly some pieces of the puzzle still missing to build those cryptosystems. We need tools that help us authenticate Alice to Bob and vice versa, and we need tools that help guarantee message integrity, allowing the receiver to verify that the received messages are in fact the messages the sender intended to send.

  1. While this might seem like an easy operation with black-and-white approximations of color mixing, keep in mind that this is just a failure of the illustration: our assumption was that this was hard. 


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