Polynomial Interpolation

If you have two points you can draw a unique line that passes through them. Suppose that you have the points $(3,3)$ and $(4,4)$. Hence, there is only one line that passes through these two points. See the plot below.

If you have three points you can draw a unique parabola that passes through them. Suppose that you have the points $(-4,16)$, $(1,1)$, and $(4,16)$. Hence, there is only one parabola that passes through these three points.

If you have four points you can draw a unique cubic polynomial that passes through them. Suppose that you have the points $(-2,8)$, $(-1,1)$, $(1,1)$, and $(2,8)$. Hence, there is only one cubic polynomial that passes through these four points.

As you might have guessed, if you have $n$ points you can draw a unique polynomial of degree $n-1$ that passes through them. This is called polynomial interpolation.

More formally, say that we have a polynomial $f(x)$ of degree $n$:

$$ f(x) = ax^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 $$

and we have $n$ points $(x_1, y_1)$, $(x_2, y_2)$, $\ldots$, $(x_n, y_n)$. Then, there is a unique polynomial $f(x)$ of degree $n-1$ such that $f(x_i) = y_i$ for $i = 1, 2, \ldots, n$.

Shamir’s Secret Sharing

Ok now let’s connect this idea to Shamir’s Secret Sharing. Suppose you encode a secret $k$ as a number. Let’s say a private key for a Bitcoin wallet. As you’ve already know, a private key is just a very big number.

You want to split this secret into $N$ parts, called shares. You also want to specify a threshold $T$ such that the secret $k$ can only be reconstructed if at least $T$ shares are combined. Here’s how you can use polynomial interpolation to achieve this.

The idea is to use polynomial interpolation to generate a polynomial $f(x)$ of degree $T-1$ such that $f(0) = k$. In other words, the polynomial $f(x)$ when evaluated at $x = 0$ should give you the secret $k$. Then, you can generate $N$ shares by evaluating $f(x)$ at $N$ different points.

Here’s an example with $T = 4$ and $N = 5$. Our secret is $k = 5$ and since $T = 4$, we generate a polynomial of degree $T-1 = 3$. We’ve chosen the polynomial $f(x) = 2x^3 - 3x^2 + 2x + 5 $. Then, we evaluate $f(x)$ at $N = 5$ different points to generate the shares.

Now this polynomial is guaranteed to pass through the point $(0, k)$. Hence if you evaluate $f(0)$ you get the secret $k$. To know the secret, you need to know the polynomial $f(x)$. And to know the polynomial $f(x)$, you need to know at least $T$ shares. Otherwise, you can’t reconstruct the polynomial and hence the secret.

In this setup we generate addresses from the extended public key (xpub) of a Bitcoin wallet that has the private key $k$. Then, we split the private key into shares and distribute them to different people. Only if at least $T$ people come together, they can reconstruct the private key and spend the funds.

Rotating Shares

Note that there’s nothing special about the points

• $(-2, f(-2))$
• $(-1, f(-1))$
• $(\frac{1}{2}, f(\frac{1}{2}))$
• $(1, f(1))$
• $(2, f(2))$

that we’ve used in the previous example. You could have chosen any other $N$ points and the polynomial would still be the same.

Suppose now that your share buddy has lost his share. Then, the participants can get together and generate a new polynomial evaluation at any point $n \notin \{ -2, -1, \frac{1}{2}, 1, 2 \}$.

This is exactly what the image below shows:

Here we’ve replaced the point $(-2, f(-2))$ with the point $(3, f(3))$. We also assume that the point $(-2, f(-2))$ is lost. The polynomial is still the same, and the secret can still be reconstructed if at least $T$ shares are combined.

We can also rotate all the shares. This is shown in the image below:

Here all previous points have been replaced by new points.

The Polynomial King

I am the Lizard Polynomial King, I can do anything!

Jim Morrison

In the end if you somehow know the polynomial $f(x)$, you can do anything. You can rug-pull all you share buddies and take all the funds.

There are several ways that a malicious actor could learn the polynomial. For example, if the shares are generated in a predictable way, an attacker could guess the polynomial. Or, during the reconstruction phase, an attacker could learn the polynomial by observing the shares. Additionally, during a distributed share generation, an attacker could disrupt the process and force the participants to reveal their shares.

Conclusion

In this post, we’ve seen how polynomial interpolation can be used to split a secret into multiple shares. We’ve also seen how the secret can be reconstructed if a certain number of shares are combined. This is the basic idea behind Shamir’s Secret Sharing (SSS).

Note that the devil is in the details. A lot of the complexities of SSS come from the details of how the shares are generated and how the secret is reconstructed. There are several types of attacks that can be done by a malicious actor. Especially during the share generation and reconstruction phases.

The intent of this blog post is to show how elegant, simple and powerful the idea behind SSS is.