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The problem with answering this question as posed is that one would need to have a definition of "cryptographic quality" that applies to 1/f noise. You'd need to talk about something like how many data points an intelligent adversary would need to observe to be able to distinguish your pseudorandom 1/f noise generator from truly random 1/f noise. I suspect ...

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For something to be a PRF, it must be impossible for a probabilistic polynomial time algorithm to distinguish it from a random function. The distinguisher you note in your attempt is correct, as a random function would not have that property (xor two calls with the same key but different x values and you get the xor of the two x values). As a side note, ...

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You have listed two properties for something to be a PRF, efficiency and pseudorandomness. The function $F'$ is clearly efficient as $F$ is efficient and $F'$ calls $F$ in just about every case. The one case where it doesn't is a very easily computable case too. So, that leaves pseudorandomness. What that property is basically saying is imagine you are ...

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As it is, there is not enough information, in particular on functions A and B, to answer. However, here are elements that may help: Is the above function using A and B a pseudorandom function as it is using LFSR to produce cipher text? As mentioned in the comments, even if the LFSR does outputs completely random numbers (which I doubt), there is ...

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HMAC is sometimes called a keyed hash. The key is not part of the input of a secure hash (as CodesInChaos already indicated in the comments). HMAC is a relatively simple construct to allow a key to be used as input to the underlying secure hash. It specifically is constructed to disallow length extension attacks - although those are not likely an issue in ...

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