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Composition of one-way functions and private keys

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lanzalibre

Are there any functions f, g and h, such that:

  1. f is one-way and is used to generate an encrypted message c when applied to message m with a public key (c = f(m, N))
  2. Original message m can only be obtained from c in polynomial time (decrypted) if a private key (k) is known
  3. A new encrypted message (c') for a desired receiver can be generated (function g) with the original encrypted message c, public key of a desired receiver N' and, if necessary, a one way function (h) of the original private keys and the public key of the desired reciver (c' = g(c, N', h(k,N'))). Extrapolation of k based on h(k,N') should not be possible in polynomial time.
  4. original message m can be extracted from c' in polynomial time (decrypted) only by the desired receiver by using his own private keys (k'), and public keys of the sender N

?

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