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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).

https://www-polsys.lip6.fr/Links/NIST/GeMSS_specification_round2.pdf

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