So if I generated a pair of (sk_1,pk) and you generated the pair (sk_2,pk), I will be able to decrypt messages that are sent to you with my secret key? ]]>

If so, then the commitment does not have to be binding, the fact that the public key is the same and the cipher text is the same does not mean that the message is the same ( I will be able to open the commitment to 2 different messages).

What do I miss here?

]]>Is it correct? I could not convince myself that it is, but the solution is dependent on it if I understood correctly.

Hope the question is clear, will write it more formally:

if G generates secret and public keys for a public key encryption, and for randomness r_g it created (sk,pk), and for randomness r_g_2 it created (sk^', pk), does it mean that sk^' == sk?

Otherwise I think there is a counter example for the solution described.

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