I am very new to Cryptography but I want to make sure I understand the concept of the scheme correctly before using it to our system. So I was looking at post quantum signature schemes, and I came across Dilithium(https://github.com/pq-crystals/dilithium), and our system currently runs on Ed25519 which based on my question can easily take a secret key and generate a deterministic public key from it.
So my question is, can Dilithium also be useable on this type of implementation, like what Elliptic Curve stuff can do? Because based on the white paper I have read in Dilithium, I just need the random 256 bit seed, and I can create the pk and sk, but I cannot take an sk and derive a pk from it. I can however get the 256 random key and generate a key from it, but it just makes the same key as the first key generation.
Let me know if this makes sense.