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  1. In IKE protocol; what is the PRF ?

  2. What is "the generation of a key based on modeled random oracle hash functions"?

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In IKE, the "PRF" is subject to negotiation between the two involved entities. There are several defined PRF in use; most are HMAC, with MD5, SHA-1 or with one of the SHA-2 functions. At least two AES-based PRF have also been defined: AES-XCBC-PRF-128 and AES-CMAC-PRF-128. The role of the PRF is to serve as internal engine for key derivation and similar usages within the protocol.

The expression:

the generation of a key based on modeled random oracle hash functions

means that a key will be generated by applying a sort-of hash function on some input, and the security of the process will be ensured if the hash function behaves like the abstract mathematical concept of a random oracle. The intuitive idea of a random oracle for key generation is that if you derive several keys from the same input set, with just minor variations (e.g. for a given input X and hash function h(), you compute h(X||1) and h(X||2) to get two derived keys), then nobody would be able to learn anything about any of the keys even if he knows all the others. The "random oracle" is the academic term which somehow means "perfect hash function".

(The mathematical details of the notion of random oracle are intricate, and I won't delve into more details here. The sentence you quote is mostly a private signal for cryptographers: it states that whoever wrote the text knows or pretends to know cryptography, and indicates to other cryptographers under what mathematical assumptions the security analysis has been performed.)

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