Identity Element on an Elliptic Curve

This section describes the 'identity element', which is the 'infinity point' in our addition and subtraction operations on an elliptic curve.

Let's take a quick break from our elliptic curve discussion to introduce the concept of "identity element" of addition and subtraction arithmetic operations on a set of elements.

The "identity element" must satisfy the following rules:

1. Any element, E, adding to the "identity element" equals to the element itself. In other words:

```E + I = E, "I" is the "identity element"
```

2. Any element, E, subtracting from the element itself equals to the "identity element".

```
E - E = I, "I" is the "identity element"
```

For example, the number zero, 0, is the "identity element" of addition and subtraction arithmetic operations over the integer set.

Now let's go back our elliptic curve discussion. You can see that "infinity point" perfectly matches the above definition of the "identity element" for addition and subtraction operations introduced on an elliptic curve.

1. Any point, P, on an elliptic curve adding to the "infinity point" equals to the point itself. In other words:

```P + O = P, "O" is the "infinity point"
```

2. Any point, P, on an elliptic curve subtracting from the point itself equals to the "infinity point".

```
P - P = O, "O" is the "infinity point"
```

Conclusion: the "infinity point" is the "identity element" of our addition and subtraction operations on an elliptic curve.

By the way, the term "identity element" is more accurate for the number 1 in multiplication and division operations over the integer set.

Maybe, the term "zero element" is more accurate for the number 0 in addition and subtraction operations over the integer set.

Last update: 2019.