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Suppose that there are Alice with attributes $A$ and $C$ and Bob with attributes $B$ and $D$. I have a ciphertext with this policy: $A\ \operatorname{AND}\ B$.

Why don't users collude ? If Alice sends the attribute $A$ to Bob, he could regenerate private key and decrypt the document. Why is this not possible ? I don't understand this concept (I'm reading Security intuition in Ciphertext-Policy Attribute-Based Encryption).

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Collusions are not possible here since each user's private keys are randomized. As you can see in the Key Generation function, for each user a random value $r$ is generated and embedded into all his private keys. Therefore, if you try to use private keys from different users, you will not be using the same random $r$ during the decryption process, so it will fail.

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  • $\begingroup$ Ok, sorry, but a user can regenerate the private key if he knows what are the attributes to satisfy the policy (for example, he discovered what are the attributes) ? These attribute are assigned from Authority or a User can select a set of attribute and generate your own private key ? $\endgroup$ – Ellipticat Mar 1 '16 at 11:34
  • $\begingroup$ Private keys are usually created by an Attribute Authority, who is in possession of a master key $MK$. Users cannot create private keys since they don't now $MK$. $\endgroup$ – cygnusv Mar 1 '16 at 11:38
  • $\begingroup$ Great. Thank you very much. Simple and excellent explanation. $\endgroup$ – Ellipticat Mar 1 '16 at 11:39
  • $\begingroup$ Sorry @cygnusv, but these attributes are assigned from Authority or a User can select a set of attributes ? $\endgroup$ – Ellipticat Mar 1 '16 at 11:53
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    $\begingroup$ That is out of the scope of the cryptographic scheme, although it is reasonable to assume that the attributes are assigned by the Attribute Authority (or by another entity trusted by it), since it will be the one generating the corresponding private keys $\endgroup$ – cygnusv Mar 1 '16 at 12:07

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