Axiom Documentation

RCM3720 Cryptography, Network and Computer Security

Enter the Rijndael field,

         8  4  3
 Z [x]/(x +x +x +x+1)
  2

and call it GR.

Determine whether x is a primitive element in this field:
- x:=generator()$GR
- primitive?(x)
Is x+1 a primitive element?
Investigate the workings of MixColumn. First create the matrix:
- M:Matrix GR:=matrix([[x,x+1,1,1],[1,x,x+1,1],[1,1,x,x+1],[x+1,1,1,x]])
Instead of multiplying a matrix C by M, we shall just look at a single column, created randomly:
- C:Matrix GR:=matrix([[random()$FR] for j in 1..4])

Remarkably enough, Axiom can operate on matrices over a finite field as easily as it can operate on numerical matrices. For example, given that
```
   D=MC 
```
it follows that
```
      -1
   C=M  D
```
or that
```
    -1
   M  D-C=0
```
To test this, first create the matrix inverse:
- MI:=inverse(M)
Now multiply by D and subtract C. What does the result tell you about the truth of the final equation?
To explore MixColumn a bit more, we shall look at the inverse of M. First, here's a small function which converts from a polynomial to an integer (treating the coefficients of the polynomial as digits of a binary number):
- poly2int(p)==(tmp:=reverse(coordinates(p)),return integer wholeRadix(tmp::LIST INT)$RadixExpansion(2))
First check the matrix M:
- map((x +-> poly2int(x)::INT), M)
Is this what you should have?
Now apply the same command but to MI instead of to M. What is the result?