Crate galois_field_2pm
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Galois Field (2M) arithmetic support.
galois_field_2pm provides wrappers for the u8, u16, u32, u64, and u128 types that implement Galois Field arithmetic. There are two implementations:
- A look up table based implementation for multiplication and division for Galois Fields defined by a primitive polynomial
- A computation based implementation for multiplication and division for Galois Fields defined by an irreducible polynomial
Getting Started
To start using a Galois Field defined by the irreducible polynomial p(x) ∈ GF(2)[x] start by representing p(x) as a u128.
An irrecuible polynomial of degree M defines the field GF(2M) = GF(2)[x] / p(x)
Next we need to choose how the field is implemented:
-
If p(x) is primitive and M ≤ 16 then the look up table implementation can be used (module gf2_lut)
-
Else use the computation based implementation (module gf2)
Lastly we must use one of the structs to represent the elements in the field. The struct GFuX can be used for M ≤ X.
At this point we can define the GF type and use it.
Consider the example below
use galois_field_2pm::{GaloisField, gf2_lut};
// First lets define out Galois Field
// For the irreducible polynomial p(x) = x^3 + x + 1 the u128 representation is given by 0xB.
// Note that this p(x) is primitive so we can use either gf2_lut of gf2
// Note that GFu8 can represent the field generate by 0xB as the degree of 0xB is 3 and 3 ≤ 8.
// For p(x) = 0x211 (degree 9), GFu8 can not be used. GFu16 must be used instead
type GF = gf2_lut::GFu8::<0xB>; // Or alternatively type GF = gf2::GFu8::<0xB>;
let a = GF::ONE;
let b = GF::new(2);
let c = GF {value: 3};
let d = (a + b * c).inverse();