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| − | '''Accompanying Notes''' | + | === '''Accompanying Notes''' === |
Finite Field/Galois Field: a finite set together with operations + and * with the following properties: | Finite Field/Galois Field: a finite set together with operations + and * with the following properties: | ||
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| − | '''A finite field exist if and only if it has size <math> p^m </math> where <math> p </math>is prime and <math> m \in \N </math> ''' | + | '''A finite field exist if and only if it has size <math> p^m </math> where <math> p </math> is prime and <math> m \in \N </math> ''' |
This is to say, there exist a Galois field with 11 elements (11 is prime, m=1) called <math> GF(11) </math> but you can not construct a Galois field with 12 elements. | This is to say, there exist a Galois field with 11 elements (11 is prime, m=1) called <math> GF(11) </math> 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''' : <math> GF(p) </math> so m = 1 and the field has a prime number of elements. | ||
| + | |||
| + | '''Extension Field ''': <math> GF(p^m) </math> 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 <math> GF(2) </math> and <math> GF(2^8) </math>. | ||
| + | |||
| + | ==== Prime Fields ==== | ||
==[[2015_Summer_Cryptography_Paar_Introduction to Galois Fields for AES_Katie Marsh_comments | Questions and comments]]== | ==[[2015_Summer_Cryptography_Paar_Introduction to Galois Fields for AES_Katie Marsh_comments | Questions and comments]]== | ||
Revision as of 05:19, 11 June 2015
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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