Key escrow on indistinguishability games Does it mean that when PPT attacker is breaking an indistinguishable based (equivalent with semantic security) game with non negligible probability that he is able to infer the secret keys either on symmetric or asymmetric encryption?
Not always, it depends on the particular encryption scheme.  Strictly speaking, the proofs only say that breaking indistinguishability is equivalent to breaking the hardness assumption they are based on.  There are some cryptosystems, like Rabin's, where the security of the key is equivalent to the security of the ciphertexts, i.e. factoring &lt;=> key recovery &lt;=> message decryption.  However, others like RSA do not have this property.  For RSA, factoring &lt;=> key recovery => message decryption, but it may be possible to decrypt messages without being able to factor.  That is why it is not said to be based on the hardness of factoring, but of the "RSA assumption".  In this situation, the security of the key is greater than or equal to that of the messages.

Long story short, the answer is generally no, but in some particular cases yes.