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You can do something like what you are suggesting. But, the EC_Dual_DBRG also has biases in the stream and so you cannot use it without changes (e.g., truncating much more). However, this is based on the same operations as ElGamal. The public key is set up exactly as proposed. Then, to encrypt a message $m$ of any length, do: Choose a random ...


1

This is - in a way - Functional Encryption. From Wikipedia: Functional encryption is a type of public-key encryption in which possessing a secret key allows one to learn a function of what the ciphertext is encrypting. So, YES, it is possible. For more information this paper might prove useful.



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