Connecting to the server, it requested me to give data that makes the last 32 bits of SHA-256 zero when added to the given prefix.
I created [a program](https://github.com/mikecat/ctf-writeups/blob/main/2021/20211218_hxp_CTF_2021/CRY/gipfel/get_S_mp.py) to solve this by brute-forcing.
(this program looks taking upto 30 minutes to get the answer on my PC. Fortunately the server was patient enough to wait for the answer.)
Dividing `g` with small values, I found that `g` modulo 4 is 1.
Therefore, I calculated the 4th root of `1` modulo `g`. One of the roots is:
Putting this value as `pubB` will make `shared` (`pubB` to the `privA`-th power modulo `g`) be `1` because `privA` is a multiple of 40, which is a multiple of 4.
Then `shared**3` and `shared**5` are also `1`.
This means that the value of `g` is printed as `verA` and we can put the value as `verB` to pass the check.
Also knowing that `g` is the SHA-256 of `password`, we can obtain the value of `password` via hashcat.
Now it is easy to obtain the encryption key `key`,
which is SHA-256 of a string created by concatenating `password`, `"\0"`, and `shared` (`1`).
Finally it's time to decrypt AES. The encryption is done with `AES.MODE_CTR` and `nonce=b''`.
CyberChef asked a 16-byte IV to decrypt in CTR mode. 16 bytes of zero worked as IV.