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In the paper Dual system encryption: realizing fully secure IBE and HIBE under simple assumption (free PDF), the authors said "there is an apparent paradox in this strategy since it seems that the reduction algorithm could simply answer the challenge ciphertext". In the paper An efficient IBE scheme with tight security reduction in the random oracle model, the authors said "Nevertheless, a private key for ID* is not helpful to decrypt challenge ciphertext, which is necessary for solving the so called self-decryption paradox".

I do not know the meaning of "self-decryption paradox" here. I think there is no restriction for the simulator to both generate the secret key for ID* and the challenge ciphertext for ID*. Thus I do not understand those points in the above papers. Could someone explain it to me?

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