# How does the public-key generation work in the multivariate post-quantum digital signature GeMMS?

There are a few steps in the public-key generation of GeMSS that I am trying to understand. The first is the below equations (1).

What does "$$\theta_i$$ forms a basis for $$\mathbb{F}_{2^n}$$ over $$\mathbb{F}_2$$" mean? I know what a basis is in linear algebra, but more details are needed so I can understand.
How do we interpret the map $$\phi$$?

1. $$(\theta_1,\ldots,\theta_n)\in(\mathbb{F}_{2^n})^n$$ form a basis for $$\mathbb{F_{2^n}}$$ over $$\mathbb{F}_2$$.
$$\phi:E=\sum_{k=1}^{n}e_k\theta_k\in\mathbb{F}_{2^n} \to \phi(E) = (e_1,\ldots,e_n)\in{\mathbb{F}_2}^n$$.

How exactly is $$f$$ created from $$F$$ in (2) below?

2) $$F=\sum_{\substack{0\leq j \lt i \lt n \\ 2^i + 2^j \leq D\\}} A_{i,j}X^{2^i+2^j} + \sum_{\substack{0\leq i \lt n \\ 2^i \leq D}} \beta_i(v_1,\ldots,v_v)X^{2^i} + \gamma(v_1,...,v_v)$$

$$f = (f_1,\ldots,f_n) \in \mathbb{F}_2[x_1,\ldots,x_{n+v}]^n$$ is created from $$F \in F_{2^n}[X,v_1,\ldots,v_v]$$ by solving the following:

$$F(\sum_{k=1}^n\theta_kx_k,v_1,\ldots,v_v) = \sum_{k=1}^{n}\theta_kf_k$$

1. The public-key is computed as the first $$m=n-\Delta$$ polynomials of $$(p_1,\ldots,p_n)=$$

$$(f_1((x_1,\ldots,x_{n+v})S),\ldots,f_n((x_1,\ldots,x_{n+v})S))T \mod \langle x_{1}^2-x_1, \ldots, x_{n+v}^2 - x_{n+v} \rangle \in \mathbb{F}_2[x_1,\ldots,x_{n+v}]^n$$

where $$(S,T)\in GL_{n+v}(\mathbb{F}_2) \times GL_n(\mathbb{F}_2)$$. What does it mean to $$\mod \langle x_{1}^2-x_1, \ldots, x_{n+v}^2 - x_{n+v} \rangle$$ by the field equations? Why are the field equations of the form $$x_{i}^2 - x_i?$$

Here is a link to the GeMMS specification for round 2 for more details (Page 6 and 7 contain key generation).