Plaintext message space in Somewhat Homormorphic Encryption scheme? Question 1: In which Somewhat Homomorphic Encryption schemes the plaintext space is defined over $\mathbb{F}_p$, where $p$ is a prime number. 

It is said, in Somewhat homomorphic encryption the noise increases when homomorphic multiplication is used. 

Question 2: Does this include multiplying a ciphertext by a plaintext? or just multiplying two ciphertexts?
Answer to question 1

I don't know about any somewhat (or leveled) homomorphic encryption whose plaintext space is the field $\mathbb{F}_p$. 

Several schemes work over the ring $\mathbb{Z}_t[x] / \langle f(x) \rangle$, which is isomorphic to the field $\mathbb{F}_{p^n}$ when we chose $t$ to be a prime $p$ and $f(x)$ to have degree $n$. However, for security reasons, we usually are not able to chose $n = 1$.

Schemes based on integers usually have the field $\mathbb{Z}_2$ as the plaintext space and they do not seem difficult to adapt to an arbitrary prime instead of $2$.

Answer to question 2

Yes, multiplying a ciphertext by a plaintext increases the noise.

Ciphertexts in these noisy schemes are (roughly speaking) of the form $f(m) + e$, where $f$ is some function, $m$ is the message, and $e$ is the error. When we multiply a ciphertext $c = f(m) + e$ by another message, let's say $m'$, we end up with something like $f(m) \cdot m' + e\cdot m' = f(m \cdot m') + e'$, and this new error $e'$ is bigger than the original error $e$.

For instance, considerer the YASHE scheme:


Ciphertext format: $c = \Delta m + e + hs$, where $h$ is the public key, $e$ and $s$ are random polynomials.
When we multiply by a plaintext $m'$: $c \cdot m' = \Delta m m' + e m' + hsm' = \Delta m m' + e' + hs'$.


As it is said on page 11 of SEAL's manual, for the FV scheme "plain multiplication increases the noise by a constant factor that depends on the plaintext multiplier".

And I am quite sure it also happens on plain multiplication of BGV scheme.