I am trying to find out if each transaction hash created by a wallet has some similarity or if every hash is "random".
I suppose my original question leads into whether or not a transaction hash be used to identify the wallet it came from?
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The procedure of generating the transaction is the same, but the value will always be pseudo random due to the nature of hash function. This post will be a good one to refer.
However if the network is public, given the transaction hash, it is possible to get the transaction details using the web3 api as documented here. It will return the address of the account where it came from.
from: String, 20 Bytes - address of the sender