Generalized Rijndael, a review

After a too long period of time with out working on this project I like to recover it.

I did a python implementation, that will be publish as free software (GPL) who is able to change the usual parameters of the Rijndael to work in a very different way than the 3 options of the standard. As a review:

Rijndael variable parameters:

• Number of rounds
• Number of rows
• Number of columns
• Wordsize (in bits)
• Number of columns in the key

The 3 standard sizes are: {10,4,4,8,4}, {12,4,4,8,6}, {14,4,4,8,8} and this means:

• Block size always 4x4 elements of 8 bits 128 bits
• Key size can have 4, 6 or 8 columns 128,192,256 bits
• And depending on this key size the number of rounds varies from 10, 12 or 14.

But this can be changed easily by something like: {40,2,2,8,8} and this means a Rijndael of 32 bits block with keys of 128 bits (doing 40 rounds, but this was set like this in the experiment because other small block ciphers have values like this).

But what means internally to the algorithm this change?

• A new irreducible polynomial is need for the mixColumn() transformation, because the polynomial ring have the same number of coefficients than the number of rows.

Are there other options in Rijndael to get this combination? Yes: this would be equivalent to {40,4,4,2,16} and what does this mean?

• A new Rijndael SBox must be build: the original is made to apply a substitution of works of 8 bits.

Pending demonstrations:

• Is the generalized Rijndael still a Pseudo-Random Permutation (PRP)
• How to build secure SBoxes?
• How to get irreducible polynomials to be able to change the number of rows?
• Is the invertible circulant matrix also valid for polynomial rings over , with ?
• How to calculate the number of rounds necessary? Not less than need to be insecure, but not more to do superfluous calculations.