Introduction to Galois Fields for AES
A slecture by student Katie Marsh
Based on the Cryptography lecture material of Prof. Paar.
Contents
Link to video on youtube
Accompanying Notes
Finite Field/Galois Field: a finite set together with operations + and * with the following properties:
1. The set forms an additive group with neutral element 0
2. The set without 0 forms a multiplicative group with neutral element 1
3. The distributive law $ a(b+c)= (ab)+(ac) $
A finite field exist if and only if it has size $ p^m $ where $ p $ is prime and $ m \in \N $
This is to say, there exist a Galois field with 11 elements (11 is prime, m=1) called $ GF(11) $ but you can not construct a Galois field with 12 elements.
There are two distinct types of Galois Fields: Prime Fields and Extension Fields
Prime Field : $ GF(p) $ so m = 1 and the field has a prime number of elements.
Extension Field : $ GF(p^m) $ where m > 1 and the field does not have a prime number of elements.
In AES, the two important fields we need to work with are $ GF(2) $ and $ GF(2^8) $.
Prime Fields
Questions and comments
If you have any questions, comments, etc. please post them here.
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