## Expand description

This crate can generate types that implement fast finite field arithmetic.

Many error correcting codes rely on some form of finite field of the form GF(2^p), where p is relatively small. Similarly some cryptographic algorithms such as AES use finite field arithmetic.

While addition and subtraction can be done quickly using just a simple XOR, multiplication is more involved. To speed things up, you can use a precomputed table. Typically this table is just copied into the source code directly.

Using this crate, you can have the benefits in speed of precomputed table, without the need to create your own type with custom multiplication and division implementation.

## WARNING

The types generated by this library are probably not suitable for cryptographic purposes, as multiplication is not guaranteed to be constant time.

## Note

The implementation was tested for finite fields up to 2^17 in size, which compiles reasonably fast. The space requirements are linear to the field size for the inversion table and log^2(N) for the multiplication table. This means it is not feasible to use this to generate fields of size 2^32, which would 4*4GB memory.

## Examples

```
g2p::g2p!(GF16, 4, modulus: 0b10011);
let one: GF16 = 1.into();
let a: GF16 = 5.into();
let b: GF16 = 4.into();
let c: GF16 = 7.into();
assert_eq!(a + c, 2.into());
assert_eq!(a - c, 2.into());
assert_eq!(a * b, c);
assert_eq!(a / c, one / b);
assert_eq!(b / b, one);
```

## Implementation details

`g2p`

generates a new type that implements all the common arithmetic operations. The
calculations are performed on either u8, u16 or u32, depending on the field size.

Addition and subtraction are implemented using regular `Xor`

. For division, the divisor inverted
using a precomputed inversion table, which is then multiplied using the multiplication outlined
below

### Multiplication

Multiplication uses a number of precomputed tables to determine the result. Because a full table would grow with the square of the field size, this approach was not deemed feasible. For example, using a full table for GF65536 = GF(2^16), one would need 2^32 entries, which would mean the program reserves 2*4GB just for these tables alone.

Instead a number `n`

is split into 8bit components `n = a + 256 * b + 65536 * c ...`

. Using this
representation we can multiply two numbers by cross-multiplying all the components
and then adding them up again. So assuming 16bit numbers `n = n0 + 256 * n1`

and
`m = m0 + 256 * m1`

we get `n*m = n0*m0 + 256*n0*m1 + 256*n1*m0 + 65536*n1*m1`

.

We can now create precomputed tables for multiplying the different components together. There is a table for first component times first component, first times second etc. The results then just have to be added together using the normal finite field addition. For our GF65536 example this means the multiplication tables use 4 * 256 * 256 entries á 2 byte which is ~0.5MB