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I am wondering about the crypto-period for keys used in pseudo-random functions.

For example, let's say I am using HMAC-SHA256 with a secret key $K$.

After how many computes do I have to update $K$?

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You don't need to change the key at all in terms of it being "overused". The only reason to change a key is when you use it beyond the bounds given in proofs of security. You have to look this up, but if you are using HMAC-SHA256 with a 256-bit key then you can go on for about $2^{128}$ computations, which you will never every do.

Having said that, the reason to replace keys is because of the fear that they may be stolen by adversaries breaching your network. This has nothing to do with crypto, but is related to other risks.

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