In the bottom of the wikipedia article of LWE (https://en.wikipedia.org/wiki/Learning_with_errors), we can see construction of Public-key cryptosystem based on the LWE.
But, I cannot understand whole thing geometrically.
From a $n$-dimensional vector space (modulo q), we sample a random vector $\mathbf{s} \in \mathbb{Z}^n_q$ which is exactly a secret key. I can easily have geometric understanding of it.
A public key is obtained by choosing $m$ vectors $\mathbf{a}_1,..., \mathbf{a}_m $ where $\mathbf{a}_i \in \mathbb{Z}^n_q$. Then, choose $m$-error offsets $e_1, ..., e_m \in \mathbb{T}$ from normal distribution. The public key is $(\mathbf{a}_i, b_i = \langle \mathbf{a}_i, \mathbf{s}\rangle / q + e_i)^{m}_{i=1}$.
Using the public key, we can encrypt a bit $x\in \{0,1\}$ by choosing a random subset S of $[m]$ and then defining Enc(x) as
$(\mathbf{a}, b) = (\sum_{i\in S} \mathbf{a}_i, \frac{x}{2} + \sum_{i \in S}b_i)$)
The decryption of $Enc(x) = (\mathbf{a}, b)$is 0 if $b - \langle \mathbf{a}, \mathbf{s} \rangle/q$ is closer to 0 than to 1/2, and 1 otherwise.
$\textbf{What I can imagine}:$
There is one secret vector (point) in $n$-dimensional vector space mod q (is $\mathbb{Z}_q^n$ a lattice)?
There are $m$-public vectors (points) in the space.
For each public vector (point), one publicly noisy constant (inner product of the vector and the secret, which is then added to an offset) $\in \mathbb{T}$ is somewhere in between points in $\mathbb{Z}_q^n$).
$\mathbf{Question}$
Q1. What is the geometric interpretation of the public key $(\mathbf{a}_i, b_i) \in (\mathbb{Z}_q^n \times \mathbb{T})^m$? What I can know is we work on $\mathbb{Z}_q^n$ and $\mathbb{T}$. But, where does exactly an element in $\mathbb{T}$ belong along with $\mathbb{Z}_q^n$? From 2 and 3 above, can we say that secret key is blurred?
Q2. What is the geometric interpretation of the ciphertext and its decryption formula? I do not know what's happening by adding public points and adding noisy constants (plus message). I guess the decryption works via inner product of $(\mathbf{a}, b)$ and $(-\mathbf{s}/q, 1) $, but why inner product?
