Adi Shamir’s Secret Sharing is an algorithm that allows participants to share ownership of a secret by distributing shares, which can be thought of as parts of a secret key. In order for someone to gain access to the original secret, a minimum number of shares (the threshold) must be used together.

Example Problem

To illustrate, let’s imagine that a family of four all share a single Bitcoin wallet. This Bitcoin wallet contains a single private key that all members of the family co-own. Because there is a single key, any of the family members can use that key to spend all of the Bitcoins.

The family has a problem: If they each keep a copy, then only one of their copies needs to be compromised to have all the coins stolen. If only one of them keeps the key, then that person may lose it or decide to double-cross the other family members.

Luckily, one of the family members is also a cryptographer. Instead of naively sharing the original key, they use SSS (Shamir’s secret sharing). The family creates four shares and sets a threshold of three, with the Bitcoin key as the original secret. Now, their plan has the following properties:

They have not stored the bitcoin key in a single place which makes it harder to steal

Members of the family need to cooperate to spend the Bitcoin, one family member can’t betray the others

If a family member dies or loses their share, the other three members can still reconstruct the key

Understanding the Threshold

Every Shamir sharing scheme has a total number of shares and a threshold. The threshold is the number of shares required to reconstruct the original secret. For example, with five shares and a threshold of three, you only need three of the five shares to calculate the original secret.

The Math – Lines

One of the fundamental mathematical properties used in Shamir’s secret sharing is the fact that it takes k points to define a polynomial of degree k – 1. For example:

Only one line can be drawn between two points

Only one possible parabola crosses through the same three points

Only one cubic curve passes through the same four points

An infinite number of lines can be drawn through the same point

An infinite number of parabolas can be drawn through the same two points

The Math – Walkthrough

Let us construct a scheme to share our secret 1954 (S) with 4 (n) shares and a threshold of 3 (k).

First, we randomly choose k – 1 positive integers, so in our case, 2 positive integers. We randomly choose 43 and 12.

Then, we build a polynomial of the form

y = a0 + a1*x + a2*x^2

Where a0 is the secret, and a1 and a2 are our randomly chosen integers. We are left with:

y = 1954 + 43x + 12x^2

Then, we use this formula to create 4 points (shares) that we give to each participant.

Share 1

(x, y) where x = 1

y = 1954 + 43*1 + 12*1^2 = 2009

(1, 2009)

Share 2

(x, y) where x = 2

y = 1954 + 43*2 + 12*2^2 = 2088

(2, 2088)

Share 3

(x, y) where x = 3

y = 1954 + 43*3 + 12*3^2 = 2191

(3, 2191)

Share 4

(x, y) where x = 4

y = 1954 + 43*4 + 12*4^2 = 2318

(4, 2318)

Reconstruction

Each participant in our scheme now owns one (x,y) point which is a single share. Remember that we set our threshold to 3 and that 3 points define a parabola (polynomial of degree 2) perfectly. That means that if we use three points, we can draw a parabola and calculate a0 (the secret).

Let’s assume we have shares 1, 2, and 4. First, we plot them:

Then we draw the corresponding parabola:

Then we find the point at x=0, whose y value is the secret:

Secret = 1954!

Note: I left out some details and restrictions in the name of simplicity, so if you want to learn more there is much more to learn on the subject.

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