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).



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.