Can you provide some context about what you are trying to achieve? I might be able to help you better.

Note that in OpenFHE, we ideally support power-of-2 cyclotomic rings, that is, the ring dimension N is a power of 2, hence, m = 2N. Non-power-of-2 cyclotomics are minimally supported and not advised to be used to instantiate any of the existing schemes in OpenFHE.

That being said, here are my answers to your questions:

You can set the ring dimension using parameters.SetRingDim(N); // => m = 2N. We do not recommend setting this parameter manually, but if you choose to do that, you may want to set the security level to HEStd_NotSet via parameters.SetSecurityLevel(HEStd_NotSet);. Note this will disable security checks of the chosen parameters, use it at your own risk.

I do not think any of the existing encoding schemes support the plaintext modulus of the form P = p^r, with p being a prime. I am afraid there is no equivalent to this. Note, that you can modify OpenFHE and add an encoding scheme that supports prime-power plaintext modulus.

You can set the bit-size of certain primes in the moduli chain via parameters.SetScalingModSize(bit_size);

It is better to get an estimate of the extended ciphertext modulus size from HElib and use the same or close size in OpenFHE. We advise to use parameters.SetKeySwitchTechnique(HYBRID);, and you can control the size of the expanded ciphertext modulus via parameters.SetNumLargeDigits(k); // k = 1, 2, 3, 4, 5 ... we recommend k = 3. You will need to dig deep in OpenFHE to find the total size of the coefficient modulus. I suggest printing the evaluation/public keys to get a feeling of what the prime moduli chain looks like.

Thank you for your insightful answers. My example application using openFHE is an overly simple key-value database. The key is in plaintext, and the value is an encrypted blob. Based on your answers, I assume I can proceed with the normal OpenFHE parameters stated in the examples, as it does several things behind the scenes.

You can use the function print_moduli_chain in this post to print all the prime factors of the extended ciphertext modulus QP in CKKS.

To control the size of the ciphertext modulus Q, you can try controling the multiplicative depth via parameters.SetMultiplicativeDepth(d); => increasing d will increase Q.

There are other ways to control P, but I assume you are not interested in that parameter.