Symmetric Variant of the BFV scheme

Is it possible to encrypt a plaintext using the secret key in SEAL?
Does the symmetric variant help in increasing the noise budget of the ciphertext, or improve the homomorphic evaluation in some other way?

No symmetric key primitives are implemented in SEAL 3.2. There are some benefits:

• Smaller initial noise;
• Possibility to replace half of a freshly encrypted ciphertext with a random seed, resulting in ~ 50% reduction in message expansion (but only in fresh ciphertexts). This can be significant.

The only problem with the symmetric key schemes is that the ciphertexts can't easily be re-randomized since without the public key there isn't any easy way to create fresh encryptions of zero. As a result, it might be hard or impossible to create provably secure protocols where the computation depends on the private data coming from other sources than the secret key owner (through multiply_plain and add_plain).