Finite field arithmetic

A finite field arithmetic is like a special kind of math where we only work with numbers that are in a set. This set is made up of a limited number of numbers that we can use, and we use them to add, subtract, multiply and divide like in regular math.

Let's imagine we have a box of toys. The box can only hold a certain amount of toys, so we can only use the toys in the box to play. In this case, instead of toys, the box is filled with numbers. These numbers might be only the numbers 0, 1, 2, and 3 or any other set of small numbers.

To add, we just put the toys together, like putting two blocks together. If we have 1 and 2 in our box, then 1 + 2 = 3.

To subtract, we take toys away. If we have 3 and 2 in our box, then 3 - 2 = 1.

To multiply, we very quickly count out groups of toys. For example, if we have two 2s and three 1s in our box, then 2 x 3 = 6 toys in total.

To divide, we have to know how many times one toy is contained in another toy. For example, if we have 6 toys in total and we want to divide them into groups of two, we have 3 groups because 6 ÷ 2 = 3.

The thing with a finite field arithmetic is that we can only use certain numbers in our box. So instead of having an endless supply of toys, we can only use a limited set.

For example, let's say our box only has the numbers 0, 1, and 2. If we add 1 and 2, we get 3. But since 3 is not in our box, we have to change it to a number that is, which is 0. This is called "wrapping around" because we get to the end of the box and start over.

So, in conclusion, finite field arithmetic is just like regular math, but we have to work with a limited set of numbers. If we add, subtract, multiply, or divide and get a number that is not in our set, we just wrap around to the beginning of the set and continue from there.