In the following proof sketch, how Eq (9) and Eq(10) are rewritten?

Question

I am trying to understand the UF-CMA and UF-NMA proof sketch of Dilithium Signature Scheme. I can't understand how the authors have used the outputs of the helping function to rewrite Eq (9) and Eq (10).

Answer 1

I have been working a fair amount with Dilithium myself, however in the version of the article I use. The equations look like this:

I do not know why the $2\gamma$ at the beginning of eq. 9 is there in your version. However, I believe the norm of u carries the higher order bits retrieved through the hint h, and that eq. 9, is then rewritten by the first bulletpoint of lemma 1, just not in great detail.

For eq. 10, I believe @Benjamin Wang is right.

I would also recommend using the Cryptography Stack Exchange community, I believe you would have a higher chance of finding someone to help you with this problem there.