I am trying to understand the paper "Breaking Symmetric Cryptosystems using Quantum Period Finding", and I reckoned the paper is roughly implying to break secret-key cryptographic system by finding the factors of the secret key. Is that correct?
Firstly, the paper is not talking about factorization at all; instead, it is using the Quantum Computer as a "constant-that-doesn't-change-the-output" algorithm (that is, find an $s$ such that $f(x) = f(x \oplus s)$ for all $x$) to break certain message authentication algorithms (which may just happen to use AES).
The paper notes that, with these specific message authentication algorithms, if you are able to find such a constant, that also gives you enough information to break the message authentication code.
Secondly, this is not actually a practical break. They run their attack in a "Quantum Oracle" model, where the attacker is given an Oracle where he can submit superpositioned queries, and get a superpositioned response. Now, in practice, this superposition state is extremely delicate; it would take rather complicated Quantum Error Correction logic to maintain it (which is actually a bit more than what we can actually implement just now). It would be quite impossible to send such a query over the internet and hope to get the correct response.
What the attacker would have to do is take this Oracle and build it in his Quantum Computer (which would implement this complication Quantum Error Correction logic), and use the attack that way. Which leads to the question: if the attacker has access to the actual implementation with the secret keys (so that he can reconstruct it within his quantum computer), why doesn't he just look inside for the keys?