IMO (and this is just my opinion) based on what I know about requirements, cookie rules and storage rules:
Performance. I have never seen the difference between the Claims and Session repositories (unless the cookies get bigger out of a lot of complaints), both of them seem to be about the same as speed (they both have to go look for data from some place (CLaims = cookie, session = server drive storage), as for the best estimate, which will fall according to how the MUCH data should be stored.
From what I saw in my experience (correct me if I am mistaken), but the session data is stored on the disk on the server and basically only have your hard disks on the hard disk for size restrictions, etc., whereas cookies they have a strict limit on the size of the encoded data and the larger number of applications that you store is more than cookies, so if you said that it maximizes this cookie, the client can see that it is amazingly performance because it sends all the data cookie in each request to the site where and in the session, the server scans the data locally, and less data is sent to the browser.
so my opinion on best practice is that your saved data to save the search in the database is a small footprint, then there really is no best practice for this, just use whatever you prefer, BUT if you store a lot of bits , especially rows in my opinion, a session would be best practice as it saves the data back and forth between the client / server and has no size limit that you can push at some point and then pull out your hair wondering why your data is missing. (he did this himself in the past, because if the cookie is too large, the client simply silently refuses it and takes 3 days to understand that this is the size of the cookie).
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