**The mathematics behind blockchain, bitcoin and NFTs.**

If you’ve ever wondered about the maths underpinning cryptocurrencies and NFTs, then here I’m going to try and work through the basic idea behind the Elliptic Curve Digital Signature Algorithm (ECDSA). Once you understand this idea you can (in theory!) create your own digital currency or NFT – complete with a digital signature that allows anyone to check that it has been issued by you.

Say I create 100 MATHSCOINS which I sell. This MATHSCOIN only has value if it can be digitally verified to be an original issued by me. To do this I share some data publicly – this then allows anyone who wants to check via its digital signature that this is a genuine MATHSCOIN. So let’s get into the maths! (Header image generated from here).

**Elliptical curves**

I will start with an elliptical curve and a chosen prime mod (here we work in mod arithmetic which is the remainder when dividing by a given mod). For this example I will be in mod 13 and my curve will be:

First I will work out all the integer solutions to this equation. For example (7,5) is a solution because:

The full set of integer solutions is given by:

Now we define addition of 2 non equal points (p_1, p_2) and (q_1, q_2) on the curve mod M by the following algorithm:

And we define the addition of 2 equal points (p_1, p_2) on the curve mod M by the following algorithm:

So in the case of (8,8) if we want to calculate (8,8) + (8,8) this gives:

This is a little tedious to do, so we can use an online generator here to calculate the full addition table of all points on the curve:

This shows that (say) (7,5) + (8,5) = (11,8) etc.

I can then chose a base point to find the order of this point (how many times it can be added to itself until it reaches the point at infinity). For example with the base point (8,8):

We can also see that the order of our starting point A(8,8) is 7 because there are 7 coordinate points (including the point at infinity) in the group when we calculate A, 2A, 3A…

**ECDSA: Elliptic Curve Signatures**

So I have chosen my curve mod M (say):

And I choose a base point on that curve (p_1, p_2) (say):

And I know the order of this base point is 7 (n=7). (n has to be prime). This gives the following:

I now chose a private key k_1:

Let’s say:

This is super-secret key. I never share this! I use this key to generate the following point on the curve:

I can see that 5(8,8) = (11,5) from my table when repeatedly adding (8,8) together.

Next I have some data z_1 which I want to give a digital signature to – this signature will show anyone who examines it that the data is authentic, has been issued by me and has not been tampered with. Let’s say:

I choose another integer k_2 such that:

Let’s say:

I am now ready to create my digital signature (s_1, s_2) by using the following algorithm:

Note, dividing by 2 is the same as multiplying by 4 in mod 7 (as this is the multiplicative inverse).

I can then release this digital signature alongside my MATHSCOIN (represented by the data z_1 = 100). Anyone can now check with me that this MATHSCOIN was really issued by me.

**Testing a digital signature**

So someone has bought a MATHSCOIN direct from me – and later on wants to sell to another buyer. Clearly this new buyer needs to check whether this is a genuine MATHSCOIN. So they have check the digital signature on the data. To allow them to do this I can share all the following data (but crucially not my private key):

This gives:

To verify someone then needs to do the following:

To verify that the data z_1 has a valid digital signature we need:

So with the shared data we have:

This verifies that the data had a valid digital signature – and that the MATHSCOIN is genuine! This is basically the foundation of all digital assets which require some stamp of authenticity.

In real life the numbers chosen are extremely large – private keys will be 256 digits long and primes very large. This makes it computationally impossible (at current speeds) to work out a private key based on public information, but still relatively easy to check a digital signature for validity.

I have also made some simple Python code which will provide a digital signature as well as check that one is valid. You can play around with these codes here:

(1) Digital signature code, (2) Checking a digital signature for validity

So time to create your own digital currency!

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