# Shamir's Secret Sharing Scheme

*This topic is an explanation of **Shamir's Secret Sharing** scheme (**SSS**) also known as \\((k, n)\\) threshold secret sharing scheme. **SSS** is one of the key parts of **RLN** due to which we can share and restore the secret.*

## Overview
Imagine, if you have some important secret (secret key) and you don't want to store it anywhere. For that you can use *SSS* scheme. It allows you to split this secret into \\(n\\) parts (each individual part doesn't give any information about the secret) and restore this secret upon presentation of \\(k\\) \\((k <= n)\\) parts.

For example, you have a secret and you want to split it into \\(n\\) parts/shares. You can divide these shares between your friends (1 share to 1 friend). Now when \\(k\\) of your friends reveal their share you can restore the secret.

This scheme is also called \\((k, n)\\) *threshold secret sharing scheme*.

This scheme is possible due to *polynomial interpolation* (especially Lagrange interpolation). Let's describe how *Lagrange interpolation* works and then how it's used in *SSS* scheme.

## Polynomial (Lagrange) interpolation

*Interpolation* is a method of constructing (or restoring) new points/values (or function) based on the range of a set of known points/values (f.e. we can restore the line (linear function) from two points, that are from this line). Previous example actually describes how that works. 
<p align="center">
    <img src="./images/graph1.png" width="300">
</p>
<p align="center">
    <i>An unlimited number of parabolas (second degree polynomials) can be drawn through two points. To choose the only one, you need a third point.</i>
</p>

Thus, if we have a polynomial \\(f(x) = 3x + 2\\) we only need two points from this polynomial to restore it. Let's peek two random \\(x\\) values and calculate \\(f(x)\\):
* For \\(x = 1\\) we have \\(f(1) = 3 * 1 + 2 = 5\\)
* For \\(x = 10\\) we have \\(f(10) = 32\\)

Now we have to shares: \\((1, 5)\\) and \\((10, 32)\\). If we draw a graph based on these two shares, we can easily see that this is the same line (function):
<p align="center">
    <img src="./images/line.png" width="500" height="400">
</p>

We also can "restore" the function analytically. For that let's denote: \\[f(x) = y_1 * \frac{x - x_2}{x_1 - x_2} + y_2 * \frac{x - x_1}{x_2 - x_1}\\]
where \\(x_1 = 5, x_2 = 10, y_1 = 5, y_2 = 32\\). If we make substitution we got: \\[f(x) = 3x + 2 \\]
which is the same polynomial.

The same techique can be made with every polynomial. Main thing to remember is that we need \\(n + 1\\) points to interpolate \\(n\\)-degree polynomial.

Now that we know how interpolation works, we can learn how it is used in SSS.

## Shamir's Secret Sharing

To create **SSS** construct we must choose \\((k, n)\\), where \\(n\\) is the number of shares we want to get from the secret and \\(k\\) is number of shares required to restore the secret. Degree of the "secret" polynomial is \\(k - 1\\) (if you don't understand why, you should reread previous part). 
Let's try to construct **SSS** with an example.

### Sharing
1. Our secret = \\(S = 30\\) 
2. As linear polynomial used in current **RLN** implementations, let's set \\(k = 2\\) (2 points are enough to recover the polynomial); \\(n\\) is not that important, but we can make it any number, f.e. 3
3. The secret polynomial is: \\[f(x) = a_1 * x + a_0 \\]
where zero coefficient \\(a_0 = S\\), and \\(a_1\\) is some random number (f.e. 5); 
4. We must pick \\(n = 3\\) different points (shares) on that polynomial, for that we can pick three random \\(x\\) values (f.e. 5, 8, 16) and calculate \\(f(x)\\):
\\[f(5) = 5 * 5 + 30 = 55\\]
\\[f(8) = 5 * 8 + 30 = 70 \\]
\\[f(16) = 5 * 16 + 30 = 110 \\]
So, the shares are: \\((5, 55), (8, 70), (16, 110)\\)

### Recovering
We can take any two shares to recover (as it was described in the interpolation section) the "secret" polynomial. Zero coefficient (\\(a_0\\)) in the recovered polynomial is the secret \\(S\\).

## Important notes
Arithmetic in this topic is usual for us. However, in the real life **SSS** arithmetic is defined over some finite field. This means that all calculations are carried out modulo some big prime number (in fact, it happens by itself in the Circom, because the arithmetic there is defined over the finite field too, so we don't need to do nothing for that). 
