**Galois Fields** $ GF(p^{n})$ - For encryption algorithms that operate on 2, 3, 4, ..., `n` bits of data at a time, we require finite fields of 2², 2³, 2⁴, 2ⁿ - The encryption algorithms used in the Advanced Encryption Standard (AES) operate on 8-bits of data at a time and require a finite field of order `256`. - This is accomplished with `256` polynomial elements that are multiplied and added modulo `m(x)`, an irreducible polynomial of degree `8`. *Example* - Fill this out from slides **Addition GF(2³) ![[Pasted image 20240131105955.png]] **Multiplication GF(2³) ![[Pasted image 20240131110012.png]]