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Nornagest comments on Harry Potter and the Methods of Rationality discussion thread, March 2015, chapter 116 - Less Wrong Discussion
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The cryptographic solution to this problem is to publicize related codes derived in such a way that the possessor of the secret code can recognize the derivation, but bystanders can't use them to rederive the secret code.
It's probably a bit much to expect Harry to use that in its strong form -- most of the relevant math was known in 1991, but it only rose to prominence with the Internet, and it's quite laborious by hand -- but there's probably a similar ad-hoc scheme he can use that'd provide reasonably strong authentication against a bunch of cryptographically naive wizards.
How much security could one expect from a mental PRNG? Simple, RNGs go back many decades so Harry could use it easily if he knew of them and thought of the application, mathematically breakable but only with knowledge of the algorithms & more samples than Harry realistically ever needs...
Does it need to be pure mental ? In some cases yes, but if he has time to carefully write himself a note, he probably has time to roll dices or write number on pieces of paper, fold them, mix them, and draw one at random. Or take a random book and look at a random letter of a random page (using some correction algorithm to deal with the difference of letter frequency).
For all practical uses x'=(x*8+1) mod 49 is a simple PRNG that can be executed mentally easily. If you seed it with the next best number you see it gives suitably random numbers for every-day purposes (and when no dice are available). Note that this is taken from TAoCP by Knuth. I use it for fair choices and mental story telling.
It's not hard to generate random numbers in your head in real life. Generate 5 or 6 "random" numbers from 0 to X-1, add them, and take the result mod X.
I don't like things which use apparatuses because they introduce a dependency (and since this scheme is for use in extreme/unusual circumstances, it's especially likely that Harry would not have leisure time or access to his pouch) and they make part of the process observable, hence, easier to realize the existence of & reverse-engineer.
A fully mental PRNG is doable under all circumstances in under a second and is unobservable except via Legilimency (which if it isn't blocked, means one is screwed anyway since one can just be False-memory-charmed into remembering having done the verification*).
* Kripkenstein would approve!
The fact that he uses prime factorization as his test for "can use you time turner to solve computationally hard problems" is evidence that he did know about prime number based cryptography, not strong evidence, but evidence still, since the prime-based crypto is the most common reason people are interested in having fast ways to factor primes.