Homomorphic Encryption and Semantic Security using Lattices?
The LWE assumption

I think we should start from the LWE assumption.
Let $n$ and $q$ be integers and let $\chi$ be a distribution over $\mathbb{Z}_q$. We often take $\chi$ as a Gaussian with small variance. (We take an error $e$ from this distribution $\chi$ and assume that $|e| \ll q$.)

The LWE assumption states that any efficient adversary cannot distinguish two oracles if a vector $\mathbf{s} \in \mathbb{Z}_q^n$ is chosen uniformly at random;


one is the random oracle that returns $(\mathbf{a},b) \in \mathbb{Z}_q^n \times \mathbb{Z}_q$ chosen uniformly at random
the other is the LWE oracle that returns $(\mathbf{a},b) \in \mathbb{Z}_q^n \times \mathbb{Z}_q$, where $\mathbf{a} \gets \mathbb{Z}_q^n$, $e \gets \chi$, and $b = \langle \mathbf{a},\mathbf{s} \rangle + 2e$.
(Note: I replace $e$ with $2e$ from the correct definition. This is a minor problem.)


The LWE-based Symmetric-key Encryption

Keeping the assumption in mind, we next review the symmetric-key encryption scheme appeared in your quotation.


$\mathsf{Gen}(n,q)$: Choose $\mathbf{s} \in \mathbb{Z}_q^n$ uniformly at random. The secret key is $\mathbf{s}$.
$\mathsf{Enc}(\mathbf{s},m \in \{0,1\})$:

Choose $\mathbf{a} \in \mathbb{Z}_q^n$ uniformly at random.
Take a sample $e$ from $\chi$.
Compute $b = \langle \mathbf{a},\mathbf{s} \rangle + 2e + m \in \mathbb{Z}_q$.
Output a ciphertext $c = (\mathbf{a},b)$.



This encryption scheme is apparently IND-CPA.
In the IND-CPA game, denoted by $G$, the adversary has an encryption oracle which has a secret key $\mathbf{s}$.
Let us replace the encryption oracle with the random oracle and call this game as $G'$.
It is easy to show that $G$ and $G'$ are indistinguishable under the LWE assumption.

Answer


  How can this be semantically secure if the same mask $\langle \mathbf{a},\mathbf{s} \rangle$ is being used on messages? That is, two different encryptions of the same value (say "10") would produce the same cipher texts.
  If I added two numbers (say "10" and "100"), wouldn't they need to be masked with the same $\langle \mathbf{a},\mathbf{s} \rangle$?
  Or does this scheme require additional accounting somewhere to allow for different $\langle \mathbf{a},\mathbf{s} \rangle$ masks?


By this definition, two ciphertexts $c_1, c_2$ of plaintexts $m_1, m_2 \in \{0,1\}$ under the secret key $\mathbf{s}$ are written as 


$c_1 = (\mathbf{a}_1, b_1) = (\mathbf{a}_1,\langle \mathbf{a}_1,\mathbf{s} \rangle + 2e_1 + m_1)$ and
$c_2 = (\mathbf{a}_2, b_2) = (\mathbf{a}_2, \langle \mathbf{a}_2,\mathbf{s} \rangle + 2e_2 + m_2)$.


If we can obtain two ciphertexts with the same mask, that is, two ciphertexts with the same randomness $\mathbf{a}$, they are not IND-CPA secure.
We have 


$c_1 = (\mathbf{a}, b_1) = (\mathbf{a},\langle \mathbf{a},\mathbf{s} \rangle + 2e_1 + m_1)$ and
$c_2 = (\mathbf{a}, b_2) = (\mathbf{a}, \langle \mathbf{a},\mathbf{s} \rangle + 2e_2 + m_2)$.


It is easy to check the relation $m_1 - m_2 \bmod{2}$ by
the relation $b_1 - b_2 \equiv 2e_1 + m_1 - (2e_2 + m_2) \bmod{q}$.
Since $e_1, e_2, m_1, m_2$ is "small", $b_1 - b_2 \bmod{q} = 2(e_1 - e_2) + m_1 - m_2 \in [-q/2,q/2]$. Therefore, the difference $b_1 - b_2 \bmod{q}$ reveals the relation $m_1 - m_2 \bmod{2}$. 

Finally, I mention that the scheme is additively homomorphic.
Suppose that we have two ciphertexts 


$c_1 = (\mathbf{a}_1, b_1) = (\mathbf{a}_1,\langle \mathbf{a}_1,\mathbf{s} \rangle + 2e_1 + m_1)$
$c_2 = (\mathbf{a}_2, b_2) = (\mathbf{a}_2, \langle \mathbf{a}_2,\mathbf{s} \rangle + 2e_2 + m_2)$.


Consider $c_1 + c_2 \in \mathbb{Z}_q^n \times \mathbb{Z}_q$.

$c_1 + c_2 = (\mathbf{a}',b')$
$ = (\mathbf{a}_1 + \mathbf{a}_2, \langle \mathbf{a}_1 + \mathbf{a}_2, \mathbf{s} \rangle + 2(e_1 + e_2 + (m_1 + m_2 \mathbin{\mathrm{div}} 2)) + (m_1 + m_2 \bmod{2}))$.

Multiplication is tricky. I recommend to read the BGV paper, which employs "tensor" to make notation simpler than the BV paper.