AES alternate equation for the S-Box affine transformation The Wikipedia article for the AES S-Box gives an alternate equation for the affine part of the S-Box transformation:

$$b_{out} = (b_{in} \times 31_d) \operatorname{mod} 257_d \oplus 99_d$$

It is not very clear in the article, but it turns out from the paper (as cited in the article) that the multiplication is to be carried out over a finite-field with polynomial $257 = 100000001_b$ for the modular reduction.

How could such an identity have been derived, since the AES specification makes no mention of the use of this other finite-field polynomial $100000001_b$? Is there any more information about where it comes from?
That alternate form, I think, makes things even more confusing than the standard form.

Since the matrix involved is circulant, the affine part of the AES S-box can be represented as 
$$
b_o = b_i \oplus (b_i \lll 1) \oplus (b_i \lll 2) \oplus (b_i \lll 3) \oplus (b_i \lll 4) \oplus 99\,,
$$
where $\oplus$ is xor and $\lll$ is bit left rotation. So far so good. But mathematically, it is awkward to describe rotation as "moving bits". 

So instead, we can treat a byte as a polynomial with coefficients modulo 2, that is, $\mathbb{F}_{2}[x]$, and a byte is $x^7\cdot c_7 + \ldots + x \cdot c_1 + c_0$, where the $c_i$ coefficients are the bits of the byte. What happens when we multiply by $x$? This is simply a left shift operation:
$$
x(x^7\cdot c_7 + \ldots + x \cdot c_1 + c_0) = x^8\cdot c_7 + \ldots + x^2 \cdot c_1 + x\cdot c_0 + 0\,.
$$
Now, if we reduce this polynomial by $x^8 + 1$, which means that we replace $x^8$ by $1$ wherever available, we get
$$
1\cdot c_7 + \ldots + x^2 \cdot c_1 + x\cdot c_0 + 0 = x^7 \cdot c_6 + \ldots + x^2 \cdot c_1 + x\cdot c_0 + c_7 \,,
$$
which is precisely a rotation left by $1$. You can verify that the same principle works for any rotation value: multiplying by $x^k$ and reducing by $x^8 + 1$ rotates the polynomial by $k$ positions.

Therefore, we can understand the affine transformation of the AES S-box as a multiplication in the ring $\mathbb{F}_{2}[x]/(x^8 + 1)$:
$$
b_o = b_i \cdot (1 + x + x^2 + x^3 + x^4) + 99 \in \mathbb{F}_{2}[x]/(x^8 + 1)\,.
$$
In other words, this modulus only exists to describe bit rotation cleanly, in an algebraic way.

Converted to decimal, this becomes $b_o = b_i \cdot 31 \bmod 257 \oplus 99 $, but that form, to me, loses all its descriptive value.