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I'm working to understand the maths behind the Rijndael cryptographic algorithm. Perhaps is too audacious to say that I understand how it works, because in fact, there are structures that I'm not sure even how they are called and what's they correct notation. I've recently ask another question and this has been pointed in the comments. I like to know how I can speak properly.

If I'm not wrong $\mathbb{F}_2$ is a field with $2$ elements (0 and 1). Two operations can be defined there, one like the addition being the xor and another like the product being the and.

It can be defined, above this binary field, an extension of some degree $\mathbb{F}_{2^n}$. Operations in this field will become what we usually do in computation bit-wise operations.

The base field can define polynomials, denoted $\mathbb{F}_{2}[x]$, but one have to define it modulo one with degree $n$ to have an equivalent to the extension mention before. Noted as: $$ \frac{\mathbb{F}_{2}[x]}{n(x)}$$

When this polynomial $n(x)$ is irreducible, it satisfy the properties of a finite field; when $n(x)$ can be factorized, it satisfies the properties of a ring.

I've mention before the Rijndael algorithm. There is a point where it defines something above those previous. I think the notation is $(\mathbb{F}_{2^n})^l$. I'm not sure if in the previous paragraphs I've already mixed concepts, but here I'm almost sure I do.

Rijndael uses a polynomial view of this structure modulo a reducible polynomial: $$\frac{\mathbb{F}_{2^n}[y]}{l(y)}$$

Then perhaps this is a quotient ring. The first notation $(\mathbb{F}_{2^n})^l$ make me think this would be a vector space but I hope not, thinking for example in the product operation that is between two elements and not by an scalar.

Finally, how can I call this last "meta"-polynomial ring that has coefficients in the binary extension, without confusing it with the polynomials defined over the binary field.

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  • $\begingroup$ Over a base field, every extension field can naturally be given the structure of a vector space (scalar multiplication is computed using the multiplication of the field). $\endgroup$ – Hurkyl Jun 30 '16 at 11:07
  • $\begingroup$ Is this "meta"-polynomial a multivariate polynomial ring? $\endgroup$ – srgblnch Jul 7 '16 at 7:50

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