MA1C ANALYTIC RECITATION 5/7/09
1.
AES
I am not sure whether you learned about vector spaces over fields other than
R
in math 1b. I thought
you might be interested to know about the Advanced Encryption Standard (AES). Notice that the set
{
0
,
1
}
under regular multiplication and addition mod 2 is a field (often called GF(2)). Therefore, we can consider
vector spaces over this field, and we can do linear algebra.
Everything you learned in 1b is still true for
linear algebra over GF(2), except stuff with eigenvalues can be weird.
Among all computers in the world doing linear algebra right now, by far the majority will be doing
linear algebra over GF(2) or another finite field like GF(2
8
). One of the reasons for this is cryptographic
computations, and in particular AES. If you know about RSA or DiffieHellman (public key cryptosystems)
you may wonder why there is a need for a regular, symmetric key cryptosystem. The reason is that public
key systems tend to be extremely costly computationally. Therefore, they are used for an initial exchange of
secret information, from which a key is generated and used in a symmetric system, like AES. The way AES
works is the following: the key is 128 bits, and data is encrypted in 128bit blocks; think of it as a four by
four array of bytes. A round of AES consists of doing an Sbox, a shiftrows, a mixcolumns, and an addkey
which are:
Sbox:
This is a nonlinear function (usually done by table lookup) on each of the 16 bytes individually.
Shiftrows:
In this step, the rows of the 4 by 4 array are shifted to the left by their row index, i.e. the
first (zeroth) row is left unchanged, the second row is shifted left one, the third twice, and the fourth
3 times.
Mixcolumns:
This is where the linear algebra comes in; each column of the array is thought of as a
vector in GF(2
8
)
4
and is multiplied by a matrix, the result of which replaces that column.
Addkey:
In the final step of the round, the key is added (remember this is mod 2, or xor) to the array
bitwise.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Ramakrishnan
 Calculus, Linear Algebra, Algebra, Vector Space, Vector field, Gradient, MA1C ANALYTIC RECITATION

Click to edit the document details