# DKGs in Groups

This is a short post on *distributed key generation* (DKG)
in the context of cryptographic groups, such as elliptic curves.

Many cryptographic schemes use a *secret key*, and a *public key*.
As their names suggest, one of them has to be kept private, and the other
can—and should be—shared.
The function these serve depends on the protocol: for signatures,
the public key is used to verify signatures, and the secret key is used
to create them; for encryption, the public key is used to encrypt,
and the secret key is used to decrypt.

Distributed key generation is a way for a group of people to generate a public key, along with shares of the secret key. These shares should be such that nobody knows the actual secret, and yet further protocols can make use of these shares to do useful operations, such as signing, or decryption.

I’ve been thinking about DKGs recently, so I wanted to get a few thoughts out there as a blog post.

# Key Generation

First, let’s look at what kind of keys we’ll be generating anyhow.

We’ll start with a cryptographic group $\mathbb{G}$, generated by $G$, of order $q$. This gives us an associated field of scalars, $\mathbb{F}_q$. We also assume that various problems related to the discrete logarithm in $\mathbb{G}$ are hard. In particular, given $A = a \cdot G$, finding $a$ should be intractable.

When working with such a group, the most common setup for keys is that a scalar $x \in \mathbb{F}_q$ is the secret key, and the associated point $X := x \cdot G$ is the corresponding public key.

Key generation here is quite simple:

- Sample $x \xleftarrow{\$} \mathbb{F}_q$.
- Return $(x, x \cdot G)$.

Now, just generating a key to use yourself isn’t all that interesting.
What *is* more interesting is generating shares of a key for multiple people.
One basic strategy here is *linear* sharing.
We pick shares $x_1, \ldots, x_n$ such that $\sum_i x_i = x$.

If we have a trusted third party, then they can be the “dealer”, and the distribute the shares $x_1, \ldots, x_n$, while informing the group of the public key $X := x \cdot G$.

What’s interesting is how to *remove* this dealer, so that a group
of people can set up a key without needing to trust a third party.

# A Broken Protocol

The way that they’ll do this will be with some kind of *protocol*.
This is a program that they’ll run together, by doing computation locally,
and exchanging messages.
The goal of the protocol will be for each party $P_i$ to end up with
their share $x_i$, along with the public key $X$, which should
satisfy $\sum_i x_i \cdot G = X$.

This protocol shouldn’t let a party learn the other shares either. Ideally, even if participants deviate from the protocol, they shouldn’t be able to learn private information or bias the result either, but we’ll get to security later.

For now, let’s just look at the basic principle behind how you’d make this work.

One simple idea is that each person can generate their share $x_i$, which implicitly creates a secret $\sum_i x_i$, shared among the participants. The “hard” part is then learning $X$ without revealing $x$, or any of the shares. This can be done by having each person send $X_i := x_i \cdot G$ to everyone else. You can then sum up these “shares” of the public key, getting $X := \sum_i X_i$.

More formally, the protocol would be:

- Each $P_i$ generates $x_i \xleftarrow{\$} \mathbb{F}_q$.
- Each $P_i$ sets $X_i \gets x_i \cdot G$.
- $\star$ Each $P_i$ sends $X_i$ to every other party.
- $\bullet$ Each $P_i$ waits to receive $X_j$ from every other $P_j$.
- Each $P_i$ sets $X \gets \sum_i X_i$.
- $\square$ Each $P_i$ returns $(x_i, X)$.

Now, because computing $x_i$ from $X_i$ is hard, this protocol is
secure, for *honest* participants.
If the parties don’t deviate from the protocol, then they won’t reveal
extra information.

### Why this is broken

However, this protocol is broken for *malicious* participants.
The reason why is that a malicious participant gains a tremendous
advantage by going “last”.
If they wait to receive the messages from other participants,
then they can actually choose the private key $x$ and the public key $X$.

After seeing $X_2, \ldots, X_n$, the malicious party can send $X - \sum_{i} X_i$, as their “share” $X_1$. This results in the final public key being $X$. If the party happens to know the logarithm of $X$, then they now control the key for the entire group, defeating the purpose of the key generation.

# A Fixed Protocol

There are two fundamental issues with the protocol so far, with regards to malicious security:

- A malicious party can choose their share $X_i$ after seeing all other $X_j$.
- A malicious party can choose thier $X_i$ without knowing its discrete logarithm.

We can fix 2. by using a Maurer proof, for the function: $$ \varphi(x) := x \cdot G $$ which will force a participant to prove that they know the discrete logarithm.

We also still need to fix 1. even with this proof, although the reason why is a bit more subtle.

Basically, even if you’re forced to know the discrete logarithm of the point
you’re submitting, it’s still possible to *bias* the resulting public key.
For example, it’s possible to try a bunch of values for your share $x_i$
until the point $X$ has a particular starting bit in its binary representation.
Can you get a useful bias in a reasonable amount of time?
Probably not.
That said, the functionality that the protocol is trying to implement
says that $X$ is completely random, so any kind of bias introduced
by a malicious participant counts as a violation of the security
properties of the protocol.

To fix this, we need to have each party set their share *before*
they learn the shares of the other parties.
We can do this using *commitments*.
A commitment hides the value underneath it, but once a commitment
is sent, the value hidden underneath can’t be changed.
We can then make each party send a commitment to their share,
and then only after receiving every other commitment will a party reveal
their own share.

Since our public shares have a lot of entropy, we can commit to them by simply using a hash function.

So, the idea now is that each person will generate $x_i$, set $X_i \gets x_i \cdot G$, and then send $H(X_i)$ to everyone else. They then wait to receive these commitments before sending out $X_i$. They then check that the received shares match the commitments, and then sum them together.

Writing this out, we get the following protocol:

**Round 1:**

- Each $P_i$ generates $x_i \xleftarrow{R} \mathbb{F}_q$, and sets $X_i \gets x_i \cdot G$.
- Each $P_i$ sets $C_i \gets H(X_i)$.
- $\star$ Each $P_i$
*broadcasts*$C_i$ to all other parties.

**Round 2:**

- $\bullet$ Each $P_i$ waits to receive $C_j$ from each other $P_j$.
- Each $P_i$ sets $\pi_i \gets \text{Prove}(\varphi, X_i; x_i)$, where: $$ \varphi(x) := x \cdot G $$
- $\star$ Each $P_i$ sends $(X_i, \pi_i)$ to all other parties.

**Round 3:**

- $\bullet$ Each $P_i$ waits to receive $(X_j, \pi_j)$ from each other $P_j$.
- $\blacktriangle$ Each $P_i$ asserts that $H(X_j) = C_j$ and that $\text{Verify}(\varphi, \pi_j, X_j)$.
- Each $P_i$ sets $X \gets \sum_{P_i \in \mathcal{P}} X_i$.
- $\square$ Each $P_i$ returns $(x_i, X)$.

And so, this is a simple protocol for doing distributed key generation with linear shares. What’s neat with linear sharing is that as soon as each person generates a random value $x_i$, they already have a sharing $x := \sum_i x_i$, and so the hard part is just sharing $X_i$ in a safe way, so that you can learn $X$.

The disadvantage of this kind of sharing is that you need *all*
of the shares in order to get the secret.
This is what we’ll remedy next.

# Threshold Key Generation

We’d like a way to share a secret in such a way that just a subset of the shares is enough to reconstruct the secret. For example, with $n$ shares, we might want to be able to use just $t < n$ shares to recreate the secret.

The most common way to do this is via polynomials. The basic property we use is that given a polynomial $f$ with $t$ coefficients, it suffices to evaluate that polynomial at $t$ points to learn the coefficients.

This reconstruction starts with a special polynomial:

$$ L_{j,S}(X) := \prod_{i \in S, i \neq j} \frac{(X - i)}{(j - i)} $$

This polynomial is such that $L_j(x) = 0$ for all $x \in [t]$
*except* $j$.
For $j$, we have $L_j(j) = 1$.
As an example, for the elements $\{0, 1, 2, 3\}$, we’d have $L_1(X)$:

We can then reconstruct our polynomial by linear combination, given enough points:

$$ f(X) = \sum_{i \in S} f(i) \cdot L_{i, S}(X) $$

Very often, we don’t care about the polynomial, but rather it’s value at $0$, which is often used as our secret. This is because $f(0)$ is just the first coefficient (depending on how you count), so it’s easy to extend a secret value into a whole polynomial, by simply adding extra random coefficients.

In case you just care about a particular value, the equation above becomes:

$$ f(0) = \sum_{i \in S} f(i) \cdot L_{i, S}(0) $$

The value $L_{i, S}(0)$ is usually written as $\lambda_i$ (with $S$ implicit), and is known as a Lagrange coefficient.

What’s interesting is that if each person has a value $f(i)$, then
they can create a linear share of $f(0)$ *locally*, by multiplying
their share $f(i)$ with $\lambda_i$.
This is convenient, because then the rest of a protocol making use
of the shares only has to work with linear values.

So, to recap, for sharing a secret $x$ among $n$ participants, such that any $t$ of them can reconstruct it, you first generate a polynomial $f$ at random, such that $f(0) = x$, and then the shares become $f(1), \ldots, f(n)$.

Given $t$ shares, you can create linear shares by multiplying with $\lambda_i$, using the set of participants $S$.

## With a Trusted Dealer

We want to distribute these shares without people actually learning the polynomial, of course.

The simplest way to do this would be to trust someone to generate this polynomial, send the shares $f(1), \ldots, f(n)$, along with $X = f(0) \cdot G$, and then forget the secret.

There are two natural flaws in this approach, one being that you need to trust the dealer to forget the secret, and the other being that the dealer might generate shares which don’t actually correspond to polynomial evaluations, and thus don’t let the participants reconstruct the secret.

# Without a Dealer

A graver flaw in using a dealer is that it’s just not fun.

Let’s remedy this with a protocol, like we did previously with just linear shares.

Instead of having a single person act as a dealer, each person
acts as a dealer.
To do this, we use the same trick with linear shares.
If each person generates a polynomial $f_i$ locally, then there’s
an *implicit* polynomial $f$ shared linearly among them, defined
as $f = \sum_i f_i$.

Another neat property is that polynomial evaluation is homomorphic, i.e. $$ f(i) + g(i) = (f + g)(i) $$

This immediately gives us a semi-honest protocol for threshold key generation:

- Each $P_i$ generates a random polynomial $f_i$ of degree $t$.
- $\star$ Each $P_i$ sends $X_i \gets f_i(0) \cdot G$ to every other party.
- $\textcolor{red}{\star}$ Each $P_i$
*privately*sends $x_i^j \gets f_i(j)$ to every other $P_j$. - $\bullet$ Each $P_i$ waits to receive $X_j$ and $x_j^i$ from every other $P_j$.
- Each $P_i$ sets $X \gets \sum_i X_i$, and $x_i \gets \sum_j x_j^i$.
- $\square$ Each $P_i$ returns $(x_i, X)$.

The idea follows the sketch from before. After generating $f_i$ locally, there’s an implicit polynomial $f$ shared linearly. Each person needs to learn $f(i)$, and $f(0) \cdot G$ is the public key. Since $f = \sum_j f_j$, it suffices to let each person act as a dealer, sending $f_j(i)$ and $f_j(0) \cdot G$, which can then be summed locally.

Of course, this only works if parties follow the protocol. If parties are malicious, we run into the same two issues we had with our original naive key generation protocol. Namely, a party can choose their inputs after seeing those of other parties, and a party can send a value $X_i$ without knowing its discrete logarithm.

Furthermore, we also run into a new issue, namely that the shares $x_i^j$ may not actually correspond to evaluations of a polynomial with the right degree, which is bad.

We’ll fix the first two with commitments and zero-knowledge proofs, as before. For the new issue, we have a bit more flexibility in how we tackle it.

# First Version

The first approach to ensuring that the shares are correct is to reveal the polynomial, but hidden behind a group element.

In essence, if you knew the coefficients of the polynomial, you would be able to check that your share was correct, via: $$ x_i^j \stackrel{?}{=} \sum_k f_{i, k} j^k $$

where $f_{i, k}$ is the $k$-th coefficient of $f_{i}$.

Now, revealing $f_{i, k}$ is obviously bad. But, revealing $f_{i, k} \cdot G$ would be ok. And if we knew $F_{i, k} = f_{i, k} \cdot G$, we could still check our equation: $$ x_i^j \cdot G \stackrel{?}{=} \sum_k j^k \cdot F_{i, k} $$

Thus, our strategy here will be to reveal these $F_{i, k}$ values, allowing others to check that our share was correctly distributed.

Naturally, we also need commitments, along with a zero-knowledge proof for $F_{i, 0}$.

In more detail, we would have:

**Round 1:**

- Each $P_i$ generates a random polynomial $f_i$ of degree $t - 1$.
- Each $P_i$ sets $F_{i, k} \gets f_{i, k} \cdot G$ for $k \in [0 \ldots t - 1]$.
- $\star$ Each $P_i$ sets $C_i \gets H(F_{i, 0})$, and then broadcasts $C_i$.

**Round 2:**

- $\bullet$ Each $P_i$ waits to receive $C_j$ from every $P_j$.
- Each $P_i$ sets $\pi_i \gets \text{Prove}(\varphi, F_{i, 0}; f_{i, 0})$, where: $$ \varphi(x) := x \cdot G $$
- $\star$ Each $P_i$ sends $F_{i, k}$ and $\pi_i$ to every other $P_j$.
- $\textcolor{red}{\star}$ Each $P_i$ privately sends $f_i(j)$ to every other $P_j$.

**Round 3:**

- $\bullet$ Each $P_i$ waits to receive $F_{j, k}$ and $\pi_j$ from every other $P_j$.
- $\blacktriangle$ Each $P_i$ asserts that $\text{Verify}(\varphi, \pi_j, F_{j, 0})$.
- $\bullet$ Each $P_i$ waits to receive $x_j^i$ from every other $P_j$.
- Each $P_i$ sets $x_i \gets \sum_j x_j^i$ and $X \gets \sum_j F_{j, 0}$.
- Each $P_i$ asserts that $x_i = \sum_k i^k \cdot \sum_j F_{j, k}$.
- $\square$ Each $P_i$ returns $(x_i, X)$.

One extra optimization is that we aggregate our verification of our share, checking the final share $x_i$ against the polynomial $f$, rather than each part $x_j^i$ against $f_j$.

One disadvantage of the aggregation is that it prevents us from identifying that a specific person gave us a bad share. If our final share is bad, we don’t know which person is at fault. You could also do an individual check, which makes attributing blame a lot easier.

Otherwise, we follow the sketch from above.

This is the most common kind of distributed key generation, at least when it comes to threshold sharings of scalars.

# Second Version

Another idea is that instead of sending $f_{i, k} \cdot G$, you instead send $F_i^j := f_i(j) \cdot G$, for each other $j$.

Then, you’ll be able to check your share $x_i^j$ via: $$ x_i^j \cdot G = F_i^j $$

Now, you do still want to check that the $F_i^j$ actually come from evaluating some polynomial $f_i$.

This can be done with a Maurer proof, using:

$$ \varphi(f) := [f(j) \cdot G | j \in [0, \ldots, n]] $$

you also want to check that they know the secret $f(0)$, so we include $j = 0$ in this list of elements as well.

Now, notice that since polynomial evaluation is homomorphic, then so will $\varphi$ be.

A sketch of the resulting protocol is then that you first generate $f_i$, then calculate $F_i^j \gets f_i(j) \cdot G$. You then send a commitment $H(F_i^0, \ldots, F_i^n)$. After seeing everyone else’s commitment, you reveal $F_i^j$, along with your proof $\pi_i$, and then privately send $f_i(j)$ to everybody else. You verify everyone else’s proof, and then sum up your private shares and the $F_j^0$ values to get the public key.

This protocol is usually worse than the other protocol we saw, but might be better for large thresholds, and where you also want to know $x_j \cdot G$, for each person’s share $x_j$. This last value is useful if you want identifiable aborts in subsequent protocols using the shares.

# Conclusion

This was just a brief note on distributed key generation, and hopefully of some use. Another good post on DKGs is “Walking Through Distributed Key Generation”, although unfortunately it uses multiplicative group notation, but this is an affront worth tolerating to read the nice introduction to the subject.

In most situations the first protocol I showed is what you’ll end up using, but I thought including the second protocol was fun, just to show that there is some flexibility in how these things are designed.